This page titled 8.1: Heat Capacity is shared under a CC BY-NC license and was authored, remixed, and/or curated by Jeremy Tatum. When the gas in vessel B is heated, it expands against the movable piston and does work \(dW = pdV\). S = standard entropy (J/mol*K) A diatomic or linear polyatomic gas has three degrees of translational freedom and two of rotational freedom, and so we would expect its molar heat capacity to be \( \frac{5}{2} RT\). {C_p} > {C_V} \ \ \ \ \ or \ \ \ \ C_{V}>C_{p} ?Cp>CVorCV>Cp? If heat is supplied at constant pressure, some of the heat supplied goes into doing external work PdV, and therefore. If we heat or do work on any gasreal or idealthe energy change is \(E=q+w\). 2 kJ b) since we're at constant pressure, H = =2.2 kJ c) H=U + (pV )= U+nRT (perfect gas) U = H nRT =2205 (3 .0 )(8 .31451)( 25) =1581 J= 1.6 kJ When a dynamic equilibrium has been established, the kinetic energy will be shared equally between each degree of translational and rotational kinetic energy. Generally, the most notable constant parameter is the volumetric heat capacity (at least for solids) which is around the value of 3 megajoule per cubic meter per kelvin:[1]. (b) When 2.0 mol CO 2 is heated at a constant pressure of 1.25 atm, its temperature increases from 250 K to 277 K. Given that the molar heat capacity of CO 2 at constant pressure is 37.11 J K 1 mol 1, calculate q, H, and U. For any ideal gas, we have, \[\frac{dE}{dT}={\left(\frac{\partial E}{\partial T}\right)}_P={\left(\frac{\partial E}{\partial T}\right)}_V=C_V \nonumber \] (one mole of any ideal gas). 12.5. But molar heat capacity at constant pressure is also temperature dependant, and the equation is . PDF (J K - Colby College The reason is that CgHg molecules are structurally more complex than CO2 molecules, and CgHg molecules have more ways to absorb added energy. Lets start with looking at Figure \(\PageIndex{1}\), which shows two vessels A and B, each containing 1 mol of the same type of ideal gas at a temperature T and a volume V. The only difference between the two vessels is that the piston at the top of A is fixed, whereas the one at the top of B is free to move against a constant external pressure p. We now consider what happens when the temperature of the gas in each vessel is slowly increased to \(T + dT\) with the addition of heat. Table 3.6. Go To: Top, Gas Phase Heat Capacity (Shomate Equation), References Data from NIST Standard Reference Database 69: NIST Chemistry WebBook The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of . *Derived data by calculation. 8: Heat Capacity, and the Expansion of Gases, { "8.01:_Heat_Capacity" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.02:_Ratio_of_the_Heat_Capacities_of_a_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.03:_Isothermal_Expansion_of_an_Ideal_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.04:_Reversible_Adiabatic_Expansion_of_an_Ideal_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.05:_The_Clement-Desormes_Experiment" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.06:_The_Slopes_of_Isotherms_and_Adiabats" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.07:_Scale_Height_in_an_Isothermal_Atmosphere" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.08:_Adiabatic_Lapse_Rate" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.09:_Numerical_Values_of_Specific_and_Molar_Heat_Capacities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "8.10:_Heat_Capacities_of_Solids" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Introductory_Remarks" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Partial_Derivatives" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Temperature" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Thermal_Conduction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Thermodynamic_Processes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Properties_of_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_The_First_and_Second_Laws_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Heat_Capacity_and_the_Expansion_of_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Enthalpy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_The_Joule_and_Joule-Thomson_Experiments" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Heat_Engines" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Free_Energy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Expansion_Compression_and_the_TdS_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_The_Clausius-Clapeyron_Equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Adiabatic_Demagnetization" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Nernst\'s_Heat_Theorem_and_the_Third_Law_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Chemical_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Experimental_Measurements" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:tatumj", "showtoc:no", "license:ccbync" ], https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FThermodynamics_and_Statistical_Mechanics%2FBook%253A_Heat_and_Thermodynamics_(Tatum)%2F08%253A_Heat_Capacity_and_the_Expansion_of_Gases%2F8.01%253A_Heat_Capacity, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 8.2: Ratio of the Heat Capacities of a Gas. A Assuming an altitude of 194 metres above mean sea level (the worldwide median altitude of human habitation), an indoor temperature of 23C, a dewpoint of 9C (40.85% relative humidity), and 760mmHg sea levelcorrected barometric pressure (molar water vapor content = 1.16%). How much heat in cal is required to raise 0.62 g of CO(g) from 316 to 396K? Ref. Thus we have to distinguish between the heat capacity at constant volume CV and the heat capacity at constant pressure CP, and, as we have seen CP > CV. A piston is compressed from a volume of 8.30 L to 2.80 L against a constant pressure of 1.90 atm. The exception we mentioned is for linear molecules. II. The molar heat capacities of real monatomic gases when well above their critical temperatures are indeed found to be close to this. In linear molecules, the moment of inertia about the internuclear axis is negligible, so there are only two degrees of rotational freedom, corresponding to rotation about two axes perpendicular to each other and to the internuclear axis. Because we want to use these properties before we get around to justifying them all, let us summarize them now: This page titled 7.13: Heat Capacities for Gases- Cv, Cp is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. b. Please read AddThis Privacy for more information. Carbon Dioxide - Thermophysical Properties - Engineering ToolBox If the heat is added at constant volume, we have simply that dU = dQ = CVdT. First let us deal with why the molar heat capacities of polyatomic molecules and some diatomic molecules are a bit higher than predicted. Each vibrational mode adds two such terms a kinetic energy term and a potential energy term. Data compilation copyright H H298.15= A*t + B*t2/2 + hXKo7h\ 0Ghrkk/ KFkz=_vfvW#JGCr8~fI+8LR\b3%,V u$HBA1f@ 5w%+@ KI4(E. where d is the number of degrees of freedom of a molecule in the system. Another way of saying this is that the energy of the collection of molecules is not affected by any interactions among the molecules; we can get the energy of the collection by adding up the energies that the individual molecules would have if they were isolated from one another. [all data], Go To: Top, Gas phase thermochemistry data, References. Molar Heat Capacities, Gases - GSU The spacing of the energy level is inversely proportional to the moment of inertia, and the moment of inertia about the internuclear axis is so small that the energy of the first rotational energy level about this axis is larger than the dissociation energy of the molecule, so indeed the molecule cannot rotate about the internuclear axis. CV = 1 n Q T with constant V. This is often expressed in the form. Quantum theory in fact accounts spectacularly well and in detail for the specific heat capacities of molecules and how the heat capacities vary with temperature. See also other properties of Carbon Dioxide at varying temperature and pressure: Density and specific weight, Dynamic and kinematic viscosity, Prandtl number, Thermal conductivity, and Thermophysical properties at standard conditions, as well as Specific heat of Air - at Constant Pressure and Varying Temperature, Air - at Constant Temperature and Varying Pressure,Ammonia, Butane, Carbon monoxide, Ethane, Ethanol, Ethylene, Hydrogen, Methane, Methanol, Nitrogen, Oxygen, Propane and Water. The heat capacity functions have a pivotal role in thermodynamics. Isotopologues: Carbon dioxide (12C16O2) Specific heat of Carbon Dioxide gas - CO2 - temperatures ranging 175 - 6000 K. Sponsored Links Carbon dioxide gas is colorless and heavier than air and has a slightly irritating odor. Please read Google Privacy & Terms for more information about how you can control adserving and the information collected. 1960 0 obj <>stream Since, for any ideal gas, \[C_V={\left(\frac{\partial E}{\partial T}\right)}_P={\left(\frac{\partial q}{\partial T}\right)}_P+{\left(\frac{\partial w}{\partial T}\right)}_P=C_P-R \nonumber \], \[C_P=C_V+R=\frac{3}{2}R+R=\frac{5}{2}R \nonumber \] (one mole of a monatomic ideal gas). Copyright for NIST Standard Reference Data is governed by I choose a gas because its volume can change very obviously on application of pressure or by changing the temperature. We don't collect information from our users. Data Program, but require an annual fee to access. Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse! boiling Constant pressure molar heat capacity of CO 2 is 37.11. However, internal energy is a state function that depends on only the temperature of an ideal gas. Vibrational energy is also quantised, but the spacing of the vibrational levels is much larger than the spacing of the rotational energy levels, so they are not excited at room temperatures. Please read Google Privacy & Terms for more information about how you can control adserving and the information collected. You can target the Engineering ToolBox by using AdWords Managed Placements. Science Chemistry The molar heat capacity at constant pressure of carbon dioxide is 29.14 J/K.mol. \[\frac{dE}{dT}={\left(\frac{\partial E}{\partial T}\right)}_P={\left(\frac{\partial E}{\partial T}\right)}_V=C_V=\frac{3}{2}R \nonumber \], It is useful to extend the idea of an ideal gas to molecules that are not monatomic. If specific heat is expressed per mole of atoms for these substances, none of the constant-volume values exceed, to any large extent, the theoretical DulongPetit limit of 25Jmol1K1 = 3R per mole of atoms (see the last column of this table). 11 JK-1mol-1 , calculate q, H and U. which of the following describes a star with a hydrogen-burning shell and an inert helium core? Some of our calculators and applications let you save application data to your local computer. When CO 2 is solved in water, the mild carbonic acid, is formed. Solved The molar heat capacity at constant pressure of - Chegg If the volume does not change, there is no overall displacement, so no work is done, and the only change in internal energy is due to the heat flow E int = Q. True, at higher temperatures the molar heat capacity does increase, though it never quite reaches \( \frac{7}{2} RT\) before the molecule dissociates. The whole-body average figure for mammals is approximately 2.9 Jcm3K1 Thus. The possibility of vibration adds more degrees of freedom, and another \( \frac{1}{2} RT\) to the molar heat capacity for each extra degree of vibration. Let us ask some further questions, which are related to these. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. B Calculated values the temperature) of the gas. Heat Capacity of a Gas - Boston University Answered: The molar heat capacity at constant | bartleby If we know an equation of state for the gas and the values of both \(C_V\) and \(C_P\), we can find the energy change between any two states of the gas, because the same change of state can be achieved in two steps, one at constant pressure and one at constant volume. Table of specific heat capacities - Wikipedia C p,solid: Constant pressure heat capacity of solid: S solid,1 bar Entropy of solid at standard conditions (1 bar) Carbon dioxide gas is colorless and heavier than air and has a slightly irritating odor. in these sites and their terms of usage. Carbon Dioxide - Specific Heat of Gas vs. Temperature - Engineering ToolBox If millions of molecules are colliding with each other, there is a constant exchange of translational and rotational kinetic energies. (I say "molar amount". why. Legal. We don't save this data. 2.3 Heat Capacity and Equipartition of Energy - OpenStax C V = 1 n Q T, with V held constant. To be strictly correct, the "number of degrees of freedom" in this connection is the number of squared terms that contribute to the internal energy. 2003-2023 Chegg Inc. All rights reserved. Indeed below about 60 K the molar heat capacity of hydrogen drops to about \( \frac{3}{2} RT\) - just as if it had become a monatomic gas or, though still diatomic, the molecules were somehow prevented from rotating. What is the change in molar enthalpy of CO2 when its temperature is increased from 298 K to 373 K at a constant pressure of 1.00 bar. The curve between the critical point and the triple point shows the carbon dioxide boiling point with changes in pressure. PDF Heat Capacities of Gases - Florida State University This page titled 3.6: Heat Capacities of an Ideal Gas is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Do they not have rotational kinetic energy?" This site is using cookies under cookie policy . We define the molar heat capacity at constant volume CV as. How do real gases behave compared with these predictions? Carbon dioxide, CO2, is a colourless and odorless gas. Answered: The molar heat capacity at constant | bartleby Its SI unit is J K1. hb```~V ce`apaiXR70tm&jJ.,Qsl,{ss_*v/=|Or`{QJ``P L@(d1v,B N`6 When CO2 is solved in water, the mild carbonic acid, is formed. Let us imagine again a gas held in a cylinder by a movable piston. Technology, Office of Data Solved When 2.0 mol CO2 is heated at a constant pressure - Chegg We define the molar heat capacity at constant volume C V as. The S.I unit of principle specific heat isJK1Kg1. %%EOF Why not? If the volume does not change, there is no overall displacement, so no work is done, and the only change in internal energy is due to the heat flow Eint = Q. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. However, NIST makes no warranties to that effect, and NIST Other names: Nitrogen gas; N2; UN 1066; UN 1977; Dinitrogen; Molecular nitrogen; Diatomic nitrogen; Nitrogen-14. dE dT = (E T)P = (E T)V = CV = 3 2R (one mole of a monatomic ideal gas) It is useful to extend the idea of an ideal gas to molecules that are not monatomic. (Recall that a gas at low pressure is nearly ideal, because then the molecules are so far apart that any intermolecular forces are negligible.) The amount of heat required to raise the temperature by one degree Celsius or one degree Kelvin when the pressure of gas is kept constant for a unit mass of gas is called principle specific heat capacity at constant pressure. It is denoted by CVC_VCV. CAS Registry Number: 7727-37-9. Answered: When 2.0 mol of CO2 is heated at a | bartleby If you want to promote your products or services in the Engineering ToolBox - please use Google Adwords. If the gas is ideal, so that there are no intermolecular forces then all of the introduced heat goes into increasing the translational kinetic energy (i.e. Carbon dioxide gas is produced from the combustion of coal or hydrocarbons or by fermentation of liquids and the breathing of humans and animals. Database and to verify that the data contained therein have Cp = heat capacity (J/mol*K) This means that if we extend our idea of ideal gases to include non-interacting polyatomic compounds, the energies of such gases still depend only on temperature. 1.50. To achieve the same increase in translational kinetic energy, the total amount of energy added must be greater. Some numerical values of specific and molar heat capacity are given in Section 8.7. been selected on the basis of sound scientific judgment. When we supply heat to (and raise the temperature of) an ideal monatomic gas, we are increasing the translational kinetic energy of the molecules. We have found \(dE_{int}\) for both an isochoric and an isobaric process. 3.5 Heat Capacities of an Ideal Gas - University Physics Volume 2 One presumes that what is meant is the specific heat capacity. Table 7.2.1: Constant Pressure Heat Capacities for a few Substances at 298.2 K and 1 bar.1 Substance He (g) Xe (g) CO (g) CO2 (g) Cp,m (J K-1 mol-1) 20.786 20.786 29.14 37.11 Substance CH4 (g) C2H6 (g, ethane) C3H8 (g, propane) C4H10 (g, n-butane) Cp,m (J K-1 mol-1) 35.309 52.63 73.51 97.45 2 Data, Monograph 9, 1998, 1-1951. 3.6: Heat Capacities of an Ideal Gas - Physics LibreTexts With volume held constant, we measure \(C_V\). Solved What is the change in molar enthalpy of CO2 when its - Chegg 5. We don't collect information from our users. Thermodynamics and Chemical Equilibrium (Ellgen), { "7.01:_Changes_in_a_State_Function_are_Independent_of_Path" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.02:_The_Total_Differential" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.03:_Line_Integrals" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.04:_Exact_Differentials_and_State_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.05:_Determining_Whether_an_Expression_is_an_Exact_Differential" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.06:_The_Chain_Rule_and_the_Divide-through_Rule" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.07:_Measuring_Pressure-Volume_Work" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.08:_Measuring_Work-_Non-Pressure-Volume_Work" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.09:_Measuring_Heat" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.10:_The_First_Law_of_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.11:_Other_Statements_of_the_First_Law" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.12:_Notation_for_Changes_in_Thermodynamic_Quantities_-_E_vs._E" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.13:_Heat_Capacities_for_Gases-_Cv_Cp" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.14:_Heat_Capacities_of_Solids-_the_Law_of_Dulong_and_Petit" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.15:_Defining_Enthalpy_H" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.16:_Heat_Transfer_in_Reversible_Processes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.17:_Free_Expansion_of_a_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.18:_Reversible_vs._Irreversible_Pressure-Volume_Work" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.19:_Isothermal_Expansions_of_An_Ideal_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.20:_Adiabatic_Expansions_of_An_Ideal_Gas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "7.21:_Problems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Introduction_-_Background_and_a_Look_Ahead" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Gas_Laws" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Distributions_Probability_and_Expected_Values" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_The_Distribution_of_Gas_Velocities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Chemical_Kinetics_Reaction_Mechanisms_and_Chemical_Equilibrium" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Equilibrium_States_and_Reversible_Processes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_State_Functions_and_The_First_Law" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Enthalpy_and_Thermochemical_Cycles" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_The_Second_Law_-_Entropy_and_Spontaneous_Change" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Some_Mathematical_Consequences_of_the_Fundamental_Equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_The_Third_Law_Absolute_Entropy_and_the_Gibbs_Free_Energy_of_Formation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Applications_of_the_Thermodynamic_Criteria_for_Change" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Equilibria_in_Reactions_of_Ideal_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Chemical_Potential_-_Extending_the_Scope_of_the_Fundamental_Equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Chemical_Potential_Fugacity_Activity_and_Equilibrium" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_The_Chemical_Activity_of_the_Components_of_a_Solution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Electrochemistry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Quantum_Mechanics_and_Molecular_Energy_Levels" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "19:_The_Distribution_of_Outcomes_for_Multiple_Trials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20:_Boltzmann_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "21:_The_Boltzmann_Distribution_Function" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "22:_Some_Basic_Applications_of_Statistical_Thermodynamics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "23:_The_Ensemble_Treatment" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "24:_Indistinguishable_Molecules_-_Statistical_Thermodynamics_of_Ideal_Gases" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "25:_Bose-Einstein_and_Fermi-Dirac_Statistics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "26:_Appendices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "showtoc:no", "license:ccbysa", "authorname:pellgen", "licenseversion:40", "source@https://www.amazon.com/Thermodynamics-Chemical-Equilibrium-Paul-Ellgen/dp/1492114278" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FThermodynamics_and_Chemical_Equilibrium_(Ellgen)%2F07%253A_State_Functions_and_The_First_Law%2F7.13%253A_Heat_Capacities_for_Gases-_Cv_Cp, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 7.12: Notation for Changes in Thermodynamic Quantities - E vs. E, 7.14: Heat Capacities of Solids- the Law of Dulong and Petit, source@https://www.amazon.com/Thermodynamics-Chemical-Equilibrium-Paul-Ellgen/dp/1492114278.