Question

With what minimum pressure (in kPa), a given volume of an ideal gas $(C_{p,m}=\frac{7}{2}R),$ originally at 400 K and 100 kPa pressure can be compressed irreversibly and adiabatically in order to raise its temperature to 600 K:

• A. 362.5 kPa
• B. 275 kPa
• C. 437.5 kPa
• D. 550 kPa

Answer 1

Answer: (B)

. 275 kPa.

Given,

$C_{p,m}=\frac{7}{2}R$

Therefore, $C_v=C_p-R=\frac{7}{2}R-R=\frac{5}{2}R$.
From the first law of thermodynamics,
$\Delta U=W+Q$
For an adiabatic irreversible compression, Q = 0.
$\therefore \Delta U=W$

The change in internal energy is given by, $\Delta U=n{C}_{v,m}(T_2-T_1)$

and the work done is given by, $W=-P_{ext}\Delta V$
$\therefore n{C}_{v,m}(T_2-T_1)=-P_{ext}(V_2-V_1)$
$n{C}_{v,m}(T_2-T_1)=-P_2[\frac{nR{T}_2}{P_2}-\frac{nR{T}_1}{P_1}]$
$\Rightarrow \frac{5}{2}R(T_2-T_1)=-P_2\times R[\frac{T_2}{P_2}-\frac{T_1}{P_1}]$
$\Rightarrow P_2=275kPa$.