Question

A piston cylinder arrangement contains an ideal gas mixture of 1 kg Nitrogen (M= 28 kg/kmol, $\gamma$ = 1.4) and 2 kg of Argon (M = 40 kg/kmol, $\gamma$ = 1.667), at 100 kPa, 300 K. The gas expands at constant pressure until the volume increases by 1 $m^3$. The heat added is approximately

• A. 145.81 kJ
• B. 97.21 kJ
• C. 291.61 kJ
• D. 218.72 kJ

Answer 1

The correct option is C 291.61 kJ

For Nitrogen,

$C_{V{N}_2}=\frac{\overline{R}}{M_{N_2}(\gamma -1)}=\frac{8.3143}{28(1.4-1)}=0.74235kJ/kgK$

For Argon,

$C_{VAr}=\frac{\overline{R}}{M_{Ar}(\gamma -1)}=\frac{8.3143}{40(1.667-1)}=0.31163kJ/kgK$

For mixture,

$C_V=\frac{m_{N_2}\times C_{V{N}_2}+m_{Ar}\times C_{VAr}}{m_{N_2}+m_{Ar}}=\frac{1\times 0.74235+2\times 0.31163}{1+2}$

$=0.4552kJ/kgK$

Equivalent gas constant mixture,

$R=\frac{m_{N_2}{R}_{N_2}+m_{Ar}+R_{Ar}}{m}=\frac{1\times \frac{8.3143}{28}+2\times \frac{8.3143}{40}}{3}$

$=\frac{0.29694+0.415715}{3}=0.23755kJ/kgK$

For mixture

$C_P=R+C_V=0.23755+0.4552$

= 0.69275 kJ/kgK

Initial volume,

$V_1=\frac{mR{T}_1}{P_1}=\frac{3\times 0.23755\times 300}{100}=2.738m^3$

Final volume,

$V_2=V_1+1=\frac{3.138}{2.138}=1.4677$

$T_2-T_1=T_1(1.4677-1)=300\times 0.4677=140.318K$

$Q=m{c}_p\times \Delta T=3\times 0.69275\times 140.318$

= 291.6158 kJ