∵ in an adiabatic process.
Q=0
So by first law of thermodynamics
Q=dU+W
⇒0=dU+W
⇒W=−dU
∴W=PdV
⇒PdV=−dU ....(i)
Given that VU∝T4⇒U=kVT4
⇒dU=kd(VT4)=K(T4dV+4T3VdT)
Also, P=31VU=31VkVT4=3KT4
Putting these values in equation (i)
⇒3KT4dV=−k(T4dV+4T3VdT)
⇒3TdV=−TdV−4VdT
⇒34TdV=−4VdT
⇒31VdV=−TdT
⇒31lnV=−lnT⇒lnV=lnT−3
⇒VT3=constant
34πR3T3=constant
RT=constant
⇒T∝R1