Question

for ideal gas the PVT equations are given by P1(V1) ^k =P2(V2) ^k and P1V1/T1 = P2V2/T2. where P, V, T are pressure, volume, temperature respectively and k is a constant show that T1/T2=(V2/V1) ^k-1​

Answer 1

Step-by-step explanation:

We know that

$P_1V_1^k = P_2V_2^k$

or

$\left(\dfrac{P_1}{P_2} \right) = \left(\dfrac{V_2}{V_1} \right)^k$ (1)

and

$\dfrac{P_1V_1}{T_1} = \dfrac{P_2V_2}{T_2}$ (2)

Let's move the P and V variables in Eqn(2) to the right side:

$\dfrac{T_1}{T_2} = \left(\dfrac{P_1}{P_2} \right)\left(\dfrac{V_1}{V_2} \right)$ (3)

Substitute Eqn(1) into Eqn(3):

$\dfrac{T_1}{T_2} = \left(\dfrac{V_2}{V_1} \right)^k \left(\dfrac{V_1}{V_2} \right)$

$\:\:\:\:\:\:\:= \left(\dfrac{V_2}{V_1} \right)^{k-1}$