The equation of state for radiant energy in equilibrium with

The equation of state for radiant energy in equilibrium with the temperature of the walls of a cavity of volume v is p = nT^3/4. the energy equation is U = aT^3V^2/4 a and b are constants. How much heat is supplied in an isothermal tripling of the volume of the cavity? dQ = dU + PdV

Solution

U = aT³V² /4

dU = aT³ *2VdV/4

dQ = (aT³ /2) VdV +bT³/4 dV

V changes from v to 3V and T is constant as the process is isothermal

Q = (aT³ /2)òVdv + bT³/4 òdv   with limits V to 3V

     = (aT³ /2) V²/2    + bT³/4 V apply limits

      = (aT³ /4) ((3V)² –V² ) + bT³/4 (3V –V)

    = 2aT³ V² + bT³V/2