(a) Show that, for an ideal gas, the relationship between molar heat capacity at
ID: 1615487 • Letter: #
Question
(a) Show that, for an ideal gas, the relationship between molar heat capacity at constant pressure, C_p, and at constant volume C_V, is C_p = C_V + R. (b) Show that for adiabatic compression or expansion of an ideal gas p^V^gamma = a, where a is a constant and gamma = C_p/C_V, and that T_2/T_1 = (V_1/V_2)^gamma - 1 (c) Prove that the work done by adiabatic expansion or compression of an ideal gas is related to the temperature change by W = nR/gamma - 1 (T_2 - T_1). (d) Calculate the final temperature and the work done when 1 mole of ammonia gas, initially at 25 degree C, is used in reversible adiabatic expansion from 0.5 L to 2.0 L. Note that C_p = 35.7 J mol^-1 K^-1. What is the change in the molar enthalpy, Delta H, and the molar internal energy, Delta U?Explanation / Answer
The heat capacities are defined as
Cp=(H/T)p Cv=(U/T)v
As we know H=U+pV
CpCv = (H/T)p(U/T)v
= (U/T)p+((pV)/T)p(U/T)v
As d(pV)=pdV+Vdp But as dp =0 therefore
CpCv=(U/T)p+p(VT)p(U/T)v
For ideal gas (V/T)p=nR/p
CpCv=(U/T)p(U/T)v +nR
As for ideal gas
(U/T)p=(U/T)v=dU/dT
Cp -Cv = nR for ideal gas at constant pressure and constant volume
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