An ideal gas experience an adiabatic compression from P = 122 kPa, V= 10.7 , m³, T= - 23.0ºC to p =1450 kPa, V= 1.36 m³. (a) Calculate the value of γ. (b) Find the final temperature. (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) Calculate the ration of the rms speed before to that after the compression.

To tackle this problem involving an ideal gas undergoing adiabatic compression, we need to apply several principles from thermodynamics. Let's break down each part of the question step by step.

Determining the Heat Capacity Ratio (γ)

The heat capacity ratio, γ (gamma), is defined as the ratio of the specific heat at constant pressure (C_p) to the specific heat at constant volume (C_v). For an ideal gas undergoing adiabatic processes, we can use the following relationship:

γ = C_p / C_v

However, we can also derive γ using the initial and final states of the gas. The adiabatic condition gives us:

• P₁V₁^γ = P₂V₂^γ

Rearranging this gives us:

γ = log(P₂/P₁) / log(V₁/V₂)

Substituting the values:

• P₁ = 122 kPa
• P₂ = 1450 kPa
• V₁ = 10.7 m³
• V₂ = 1.36 m³

Calculating the ratio:

γ = log(1450/122) / log(10.7/1.36)

After performing the calculations, we find:

γ ≈ 1.4

Finding the Final Temperature

To find the final temperature (T₂), we can use the adiabatic relation:

T₁V₁^(γ-1) = T₂V₂^(γ-1)

First, convert T₁ from Celsius to Kelvin:

T₁ = -23.0 + 273.15 = 250.15 K

Now substituting the known values:

250.15 * (10.7)^(1.4 - 1) = T₂ * (1.36)^(1.4 - 1)

Solving for T₂ gives:

T₂ ≈ 1000 K

Calculating the Number of Moles

To find the number of moles (n), we can use the ideal gas law:

PV = nRT

Using the initial conditions:

• P = 122 kPa = 122,000 Pa
• V = 10.7 m³
• R = 8.314 J/(mol·K)
• T = 250.15 K

Rearranging the ideal gas law gives:

n = PV / RT

Substituting the values:

n = (122,000 * 10.7) / (8.314 * 250.15)

Calculating this yields:

n ≈ 6.0 moles

Translational Kinetic Energy per Mole

The translational kinetic energy (KE) per mole of an ideal gas can be expressed as:

KE = (3/2)RT

Calculating for both states:

• Before compression (T₁): KE₁ = (3/2) * 8.314 * 250.15
• After compression (T₂): KE₂ = (3/2) * 8.314 * 1000

Calculating these values gives:

• KE₁ ≈ 3117.5 J/mol
• KE₂ ≈ 12456 J/mol

RMS Speed Ratio

The root mean square (rms) speed of gas molecules is given by:

v_rms = √(3RT/M)

To find the ratio of the rms speed before and after compression:

v_rms,1 / v_rms,2 = √(T₁ / T₂)

Substituting the temperatures:

v_rms,1 / v_rms,2 = √(250.15 / 1000)

This simplifies to:

v_rms,1 / v_rms,2 ≈ 0.5

In summary, we have calculated the heat capacity ratio, final temperature, number of moles, translational kinetic energy per mole, and the ratio of rms speeds before and after the compression. Each step utilized fundamental thermodynamic principles and the ideal gas law to arrive at the answers.