Question

For an ideal gas, the molar specific heat capacities at constant pressure and volume are $C_p$ and $C_V$ respectively. Choose the correct option.

• A. $C_V={\left(\frac{ΔU}{ΔT}\right)}_V$
• B. $C_p={\left(\frac{ΔU}{ΔT}\right)}_p+p{\left(\frac{ΔV}{ΔT}\right)}_p$
• C. $p{\left(\frac{ΔV}{ΔT}\right)}_p=R$
• D. All of these

Answer 1

Answer: (D)

Using first law of thermodynamics,
$ΔQ=ΔU+pΔV$
At constant $V,ΔV=0$
$C_V={\left(\frac{ΔQ}{ΔT}\right)}_V={\left(\frac{ΔU}{ΔT}\right)}_V$
At constant pressure $p,Δp=0$
$C_p={\left(\frac{ΔQ}{ΔT}\right)}_p={\left(\frac{ΔU}{ΔT}\right)}_p+{\left(\frac{pΔV}{ΔT}\right)}_p$
$={\left(\frac{ΔU}{ΔT}\right)}_p+p{\left(\frac{ΔV}{ΔT}\right)}_p$
From ideal gas equation,
$pV=μRT$
For $μ=1,pV=RT$
At constant pressure $p,$
$pΔV=RΔT$
$p{\left(\frac{ΔV}{ΔT}\right)}_p=R$
Thus, all the equations given in the options are correct for an ideal gas.