An ideal diatomic gas is initially at a temperature T = 0 degree celcius . Then it expands reversibly and adiabatically to 5 times its volume . Its final temperature will be approximately

To determine the final temperature of an ideal diatomic gas that expands reversibly and adiabatically to five times its initial volume, we can use the principles of thermodynamics, specifically the relationships governing adiabatic processes. Let's break this down step by step.

Understanding Adiabatic Expansion

In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas, the relationship between temperature and volume during an adiabatic expansion can be described by the equation:

T1 * V1^(γ - 1) = T2 * V2^(γ - 1)

Here, T1 and T2 are the initial and final temperatures, V1 and V2 are the initial and final volumes, and γ (gamma) is the heat capacity ratio, which for a diatomic gas is approximately 1.4.

Given Values

• Initial temperature (T1) = 0 °C = 273 K
• Initial volume (V1) = V
• Final volume (V2) = 5V
• γ for diatomic gas = 1.4

Applying the Adiabatic Relation

We can substitute the known values into the adiabatic equation:

273 K * V^(1.4 - 1) = T2 * (5V)^(1.4 - 1)

This simplifies to:

273 K * V^(0.4) = T2 * (5^(0.4) * V^(0.4))

Notice that the V^(0.4) terms cancel out:

273 K = T2 * 5^(0.4)

Calculating the Final Temperature

Now, we need to calculate 5^(0.4). This value is approximately 2.297. Plugging this back into our equation gives:

273 K = T2 * 2.297

To find T2, we rearrange the equation:

T2 = 273 K / 2.297

Calculating this gives:

T2 ≈ 118.5 K

Converting Back to Celsius

Finally, to express the final temperature in degrees Celsius, we convert from Kelvin:

T2 (°C) = T2 (K) - 273

T2 (°C) ≈ 118.5 K - 273 ≈ -154.5 °C

Thus, the final temperature of the gas after it expands adiabatically to five times its initial volume is approximately -154.5 °C. This significant drop in temperature is characteristic of adiabatic expansion, where the gas does work on its surroundings without heat input, leading to a decrease in internal energy and, consequently, temperature.