An ideal gas, for which gamma=1.5, undergoes an adiabatic expansion in which its volume is doubled. if the initial temp if the gas is 27 C, what is the final temperature of the gas?

To solve this problem, we can use the adiabatic expansion formula for temperature changes in an ideal gas:

(T2 / T1) = (V1 / V2)^(gamma-1)

where:
T1 = initial temperature of the gas
T2 = final temperature of the gas
V1 = initial volume of the gas
V2 = final volume of the gas
gamma = specific heat ratio for the gas

Here, we are given:
gamma = 1.5
V2 = 2 * V1 (since the volume is doubled)
T1 = 27 °C = 300 K (converted to Kelvin)

Now, let's plug in the values and solve for T2:

(T2 / 300) = (V1 / (2 * V1))^(1.5 - 1)
(T2 / 300) = (1 / 2)^(0.5)
(T2 / 300) = 0.707

To find T2, multiply both sides by 300:

T2 = 0.707 * 300
T2 = 212.1 K

Therefore, the final temperature of the gas is approximately 212.1 K.