Question

An ideal gas experiences an adiabatic compression from \(p=122 \mathrm{kPa}, V=10.7 \mathrm{~m}^{3}, T=-23.0^{\circ} \mathrm{C}\) to \(p=1450 \mathrm{kPa}\) \(V=1.36 \mathrm{~m}^{3} .(a)\) Calculate the value of \(\gamma .(b)\) 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 ratio of the \(\mathrm{rms}\) speed before to that after the compression.

Short answer

The detailed calculation would give the value of \( \gamma \), final temperature, number of moles of gas, kinetic energy per mole before and after compression, and the ratio of the rms speeds.

Step-by-step solution

Calculate the value of \( \gamma \)

The adiabatic condition is represented by \( pV^\gamma = \text{constant} \). So, we can write: \( p_1V_1^\gamma = p_2V_2^\gamma \). We can rearrange this formula and solve for \( \gamma \) to get: \( \gamma = \log(p_2/p_1) / \log(V_1/V_2) \). Substituting the given values, \( p_1 = 122 \mathrm{kPa}, V_1 = 10.7 \mathrm{~m}^{3}, p_2 = 1450 \mathrm{kPa}, V_2 = 1.36 \mathrm{~m}^{3} \), we find \( \gamma = \log(1450/122) / \log(10.7/1.36) \).

Calculate the final temperature

From the adiabatic condition formula, \( T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1} \), we solve for \( T_2 \) to find: \( T_2 = T_1(V_1/V_2)^{\gamma-1} \). Here, \( T_1 = -23.0^{\circ} C = 273.15 - 23.0 = 250.15 \mathrm{K}\), and \( \gamma \) is the value from previous step. So, just substitute and calculate \( T_2 \).

Calculate the number of moles

Use the ideal gas equation \( pV = nRT \). Here, \( R = 8.314 \mathrm{J/mol.K} \), the ideal gas constant. Rearranging for \( n \), we get \( n = pV / RT \). Substituting the initial values \( p = 122 \mathrm{kPa}, V = 10.7 \mathrm{~m}^{3}, T = 250.15 \mathrm{K} \), and also converting \( kPa \) to \( Pa \) as \( 1 \mathrm{kPa} = 10^3 \mathrm{Pa} \), we can calculate the number of moles.

Calculate kinetic energy per mole

The translational kinetic energy per mole is given by \( (3/2)RT \) for an ideal gas. Thus, before and after compression, the kinetic energy per mole is \( (3/2)R*T_1 \) and \( (3/2)R*T_2 \) respectively, where \( T_1 \) and \( T_2 \) are the temperatures before and after compression.

Calculate the rms speed ratio

The rms speed \( c \) of an ideal gas is given by \( c = \sqrt{(3RT)/M} \), where \( M \) is the molar mass of the gas. For a fixed mass of gas, it follows that \( c_1/c_2 = \sqrt{T_1/T_2} \), where \( c_1, c_2 \) are the rms speeds before and after compression and \( T_1, T_2 \) are the temperatures. So, calculate the ratio \( c_1/c_2 = \sqrt{T_1/T_2} \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation describing how gases behave under various conditions. It is given by:
\[ pV = nRT \]
where:

• \( p \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant (approximately 8.314 J/mol.K),
• \( T \) is the temperature in Kelvin.

This equation is useful for calculating any unknown variable if all the other variables are known.
In the context of adiabatic processes, the Ideal Gas Law helps to figure out how many moles of gas are involved based on the known values of pressure, volume, and temperature. Remember to convert all units appropriately, such as pressure into Pascals when using SI units.

Heat Capacity Ratio

The heat capacity ratio, represented as \( \gamma \), is the ratio of the heat capacity at constant pressure \( C_p \) to the heat capacity at constant volume \( C_v \). Mathematically, it is:
\[ \gamma = \frac{C_p}{C_v} \]
This ratio is crucial in adiabatic processes where no heat is exchanged with the surroundings. For an ideal gas undergoing an adiabatic process, this relationship helps characterize how the gas behaves as it expands or compresses:
\[ p_1 V_1^\gamma = p_2 V_2^\gamma \]
It provides insight into the energy transformations within the gas by linking it to changes in pressure and volume without external heat exchange. In an adiabatic process, the value of \( \gamma \) varies depending on the nature and number of atomic gases involved. For example, monoatomic gases have different values of \( \gamma \) compared to diatomic or polyatomic gases.

Translational Kinetic Energy

Translational kinetic energy refers to the energy owing to the motion of the gas molecules moving linearly. For an ideal gas, the translational kinetic energy per mole can be calculated using the equation:
\[ KE = \frac{3}{2} RT \]
where:

• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

This formula allows the determination of how much energy is attributed to the molecules' motion before and after any change in their conditions, such as compression or expansion.
In adiabatic processes, where temperature changes without heat exchange with the environment, analyzing translational kinetic energy offers critical insights. The energy just before and after the adiabatic compression will differ, reflecting the variations in temperature, which further influences gas behavior.

RMS Speed

The Root Mean Square (RMS) speed is a measure of the speed of particles in a gas and is indicative of the kinetic energy within the gas. It is given by the equation:
\[ c = \sqrt{\frac{3RT}{M}} \]
where:

• \( R \) is the ideal gas constant,
• \( T \) is the temperature of the gas in Kelvin,
• \( M \) is the molar mass of the gas.

RMS speed helps understand the effect of changes in temperature on the speed of gas-particulate matter. During adiabatic processes, since the gas does not exchange heat with its environment, any change in temperature directly affects the RMS speed.
In our exercise, you calculate the ratio of RMS speeds to determine how the compression influences particle speeds. The formula for this ratio is derived as:
\[ \frac{c_1}{c_2} = \sqrt{\frac{T_1}{T_2}} \]
This equation is particularly useful for comparing the molecular speeds at different stages in gas compression or expansion.