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Asked by Damanpreet Last Modified
Answer The difference between the molar specific heats at constant pressure (CpCp) and constant volume (CvCv) for an ideal gas can be derived using the relationship between enthalpy (HH), internal energy (UU), pressure (pp), and volume (VV):
H=U+pVH=U+pV
For an ideal gas, the internal energy only depends on temperature and the number of moles (nn):
U=nCvTU=nCvT
And, for an ideal gas, the enthalpy depends on the internal energy, pressure, and volume:
H=nCpTH=nCpT
Using these relationships and the given empirical relation, we can derive the difference between CpCp and CvCv:
H=U+pVH=U+pV
nCpT=nCvT+pnRTnCpT=nCvT+pnRT
Dividing both sides by nTnT:
Cp=Cv+pVnTCp=Cv+nTpV
For an ideal gas, we know that pV=nRTpV=nRT, so:
Cp=Cv+nRTnTCp=Cv+nTnRT
Cp=Cv+RCp=Cv+R
Therefore, the difference between CpCp and CvCv for an ideal gas is equal to the universal gas constant (RR).
Given that we're considering 10 moles of an ideal gas, we'll use the value of the universal gas constant (RR) for the calculation. The value of RR depends on the units used, but commonly it's approximately 8.314 J/mol⋅K8.314J/mol⋅K.
Difference between Cp and Cv=RDifference between Cp and Cv=R
Difference between Cp and Cv=10 moles×8.314 J/mol⋅KDifference between Cp and Cv=10moles×8.314J/mol⋅K
Difference between Cp and Cv≈83.14 J/KDifference between Cp and Cv≈83.14J/K
So, for 10 moles of an ideal gas, the difference between CpCp and CvCv is approximately 83.14 J/K83.14J/K.
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