Question

Two systems have the following equations of state, where R is the gas constant per mole. The mole numbers are N1 = 3 and N2 = 5. The initial temperatures are T1 = 175 K and T2 = 400 K. What is the temperature once equilibrium is reached (in Kelvin)?

Answer 1

we know that, for two particular system we have,$T_1=175 K,T_2=400K$

$n_1=3,n_2=5,$

we know that

$\frac{1}{T}=\frac{3}{2}R (\frac{n}{U})$

for two different temperature

$\frac{1}{T_1}=\frac{3}{2}R (\frac{n_1}{U_1})$

$\frac{1}{175}=\frac{3}{2}R (\frac{3}{U_1})$

$U_1=6547.275$

similarly for other system

$\frac{1}{T_2}=\frac{3}{2}R (\frac{n_2}{U_2})$

$\frac{1}{400}=\frac{3}{2}R (\frac{5}{U_2})$

$U_2=24942 J$

so,

$U=U_2-U_1=24942-6547.275$

so, during equilibrium condition

$\frac{1}{T}=\frac{3}{2}R (\frac{n}{U})$

$\frac{1}{T}=\frac{3}{2}(8.314)(\frac{8}{18394.725})$

T= 184.375 K

This is the equilibrium temperature