A rigid wellinsulated tank contains two ideal gases separate

A rigid, well-insulated tank contains two ideal gases, separated by piston held in place by a pin. Initially, on one side of the piston is 1 kg of N_2 at 500 kPa and 350 K and on the other side of the piston is 3 kg of CO at 200 kPa and 450 K. The pin is removed and the final state is deemed when the two gases reach thermodynamic equilibrium. Neglect changes in potential and kinetic energies and the mass of the piston. Assume constant specific heat for each substance. Determine the final equilibrium temperature. Is the constant specific heat assumption justified? Demonstrate with numbers.

Solution

Here,

Tn = temperature of nitrogen = 350K

Tc = temperature of CO = 450 K

Cn = specific heat of nitrogen = 1.041 kJ/kg-K

Cc = specific heat of CO = 1.054 kJ/kg-K

Let\'s assume Cn and Cc are constant

Mn = mass of nitrogen = 1 kg

Mc = mass of CO = 3 kg

Tf = Final equilibrium temperature

(a) As the system is insulated i.e. adiabatic, so total heat transfer by nitrogen is equal to that of CO.

So,

Mn x Cn x (Tf - Tn) = Mc x Cc x (Tc - Tf)

or, 1 x 1.041 x (Tf - 350) = 3 x 1.054 x (450 - Tf)   

So, Tf = 425.28 K

The final equilibrium temperature is 425.28 K.

(b) The assumption is justified. Because for the range of temperature present here, the specific doesn\'t change much.

Nitrogen has scecific heat of 1.047 kJ/kg-K at 425K and 1.041 kJ/kg-K at 350K. CO has scecific heat of 1.051 kJ/kg-K at 425K and 1.054 kJ/kg-K at 450K.

From these data, we can conclude that the constant specific heat assumption is justified.