Question

Two mole of ideal diatomic gas $(C_{V,m}=5/2R)$ at 300 K and 5 atm expanded irreversibly and adiabatically to a final pressure of 2 atm against a constant pressure of 1 atm.Calculate q, w, $\Delta H$ and $\Delta U$.

Answer 1

Given that:-

$P_{ext.}=1atm$

$P_1=5atm$

$P_2=2atm$

$T_1=300K$

$T_2=T\left(\text{say}\right)=?$

$n=2$

$C_V=\frac{5}{2}R$

For an adiabatic process,

$W=n{C}_{V}dT=-P\Delta V$

$\therefore n{C}_V(T_2-T_1)=-P_{ext}(V_2-V_1)$

$\Rightarrow 2\times \frac{5}{2}R(T-300)=-1(\frac{nR{T}_2}{P_2}-\frac{nR{T}_1}{P_1})$

$\Rightarrow 5R(T-300)=-2R(\frac{T}{2}-\frac{300}{5})$

$\Rightarrow 5T-1500=-T+120$

$\Rightarrow 6T=1620\Rightarrow T=270K$

$\therefore T_2=T=270K$

$\therefore W=n{C}_{V}dT=2\times \frac{5}{2}R\times (270-300)=-1247.1J$

$\Rightarrow q=0$

$\Delta U=W=-1247.1J$

$\therefore \Delta H=\Delta U+nR\Delta T$

$\Rightarrow \Delta H=-1247.1+2\times 8.314\times (270-300)=-1745.94J$