2 mole of H2 gas are compressed adiabitacily from STP condition to occupy a volume of 4.48 litre calculate tha final temperature

To find the final temperature of 2 moles of hydrogen gas (H2) that are compressed adiabatically from standard temperature and pressure (STP) to occupy a volume of 4.48 liters, we can use the principles of thermodynamics, specifically the adiabatic process equations. Let's break this down step by step.

Understanding the Initial Conditions

At STP, the conditions are defined as:

• Temperature (T1) = 273.15 K (0 °C)
• Pressure (P1) = 1 atm
• Volume (V1) = 22.4 L for 1 mole of an ideal gas

Since we have 2 moles of H2, the initial volume (V1) at STP would be:

V1 = 2 moles × 22.4 L/mole = 44.8 L

Applying the Adiabatic Process Formula

In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure, volume, and temperature for an ideal gas undergoing an adiabatic process can be expressed as:

P1V1^γ = P2V2^γ

and

T1V1^(γ-1) = T2V2^(γ-1)

where γ (gamma) is the heat capacity ratio (C_p/C_v). For diatomic gases like H2, γ is approximately 1.4.

Calculating the Final Temperature

We know:

• T1 = 273.15 K
• V1 = 44.8 L
• V2 = 4.48 L

Now we can rearrange the equation to solve for T2:

T2 = T1 × (V1 / V2)^(γ - 1)

Substituting the known values:

T2 = 273.15 K × (44.8 L / 4.48 L)^(1.4 - 1)

Calculating the volume ratio:

V1 / V2 = 44.8 / 4.48 = 10

Now substituting this back into the equation:

T2 = 273.15 K × (10)^(0.4)

Calculating (10)^(0.4):

(10)^(0.4) ≈ 2.5119

Finally, we calculate T2:

T2 ≈ 273.15 K × 2.5119 ≈ 686.6 K

Final Result

The final temperature of the hydrogen gas after adiabatic compression to a volume of 4.48 liters is approximately 686.6 K.

This example illustrates how the principles of thermodynamics apply to real-world scenarios, such as gas compression, and highlights the importance of understanding the relationships between pressure, volume, and temperature in different processes.