One mole of an ideal gas expands adiabatically from w an initial state (𝑇A, 𝑉0) to final state (𝑇f , 5𝑉0). Another mole of the same gas expands isothermally from a different initial state (𝑇B,𝑉0) to the same final state (𝑇f , 5𝑉0). The ratio of the specific heats at constant pressure and constant volume of this ideal gas is 𝛾. What is the ratio 𝑇A/𝑇B? (A) 5 𝛾−1 (B) 5 1−𝛾 (C) 5 𝛾 (D) 5 1+�

 To find the ratio 

$\frac{T_A}{T_B}$, we analyze the two processes described:

1. Adiabatic Process:

   • The gas expands adiabatically from $(T_A,V_0)$ to $(T_f,5V_0)$.

   • For an adiabatic process, the relation between temperature and volume is given by:

     $$T_A{V}_0^{\gamma -1}=T_f(5V_0{)}^{\gamma -1}$$

   • Solving for $T_A$:

     $$T_A=T_f{\left(\frac{5V_0}{V_0}\right)}^{\gamma -1}=T_f\cdot 5^{\gamma -1}$$

2. Isothermal Process:

   • The gas expands isothermally from $(T_B,V_0)$ to $(T_f,5V_0)$.

   • For an isothermal process, the temperature remains constant, so $T_B=T_f$.

3. Ratio Calculation:

   • From the adiabatic process, we have $T_A=T_f\cdot 5^{\gamma -1}$.

   • From the isothermal process, we have $T_B=T_f$.

   • Therefore, the ratio $\frac{T_A}{T_B}$ is:

     $$\frac{T_A}{T_B}=\frac{T_f\cdot 5^{\gamma -1}}{T_f}=5^{\gamma -1}$$

Thus, the final answer is $\boxed{{\displaystyle A}}$.