Question

\((a)\) A monatomic ideal gas initially at \(19.0^{\circ} \mathrm{C}\) is suddenly compressed to one-tenth its original volume. What is its temperature after compression? (b) Make the same calculation for a diatomic gas.

Short answer

The final temperature for a monatomic gas is approximately 493 K. For a diatomic gas, it's approximately 553 K.

Step-by-step solution

Convert Temperature to Kelvin

The initial temperature is given in Celsius. It needs to be converted to Kelvin, as this is the unit of temperature in the International System of Units (SI). The conversion can be done using the formula \(T(K) = T(C) + 273.15\). Thus, \(T_0 = 19.0C + 273.15 = 292.15 K\).

Calculate the Final Temperature for a Monatomic Gas

The adiabatic equation \(T_{0}V_{0}^{\gamma - 1} = T_{1}V_{1}^{\gamma - 1}\) should be used to find the final temperature for a monatomic gas. Given the initial volume \(V_0\) is ten times the final volume \(V_1\), we can write the equation as \(T_{0}10^{\gamma - 1} = T_{1}\). Solving this equation for \(T_1\) yields \(T_1 = T_{0}10^{\gamma - 1}\). For a monatomic gas, \(\gamma = 5/3\). Substituting \(\gamma\) and \(T_0 = 292.15K\) into the equation yields the final temperature \(T_{1} = 292.15 K \times 10^{5/3 - 1} \approx 493 K\).

Calculate the Final Temperature for a Diatomic Gas

Use the same process for a diatomic gas. Only the value of \(\gamma\) changes to \(7/5\) in case of a diatomic gas. Substituting \(\gamma = 7/5\) and \(T_0 = 292.15K\) into the equation yields \(T_{1} = 292.15 K \times 10^{7/5 - 1}\), or approximately \(T_{1} \approx 553 K\).

Key concepts

Monatomic Ideal Gas

A monatomic ideal gas is a simple conceptual model used in thermodynamics. It consists of gas particles that are single atoms, like helium or neon. These atoms have no molecular bonds and move freely.
Because the gas is monatomic, it has a smaller heat capacity compared to complex gases.

• Fluctuations in this type of gas are predictable when undergoing processes like adiabatic compression.
• The adiabatic index, represented as \( \gamma \), is calculated as \( \frac{5}{3} \) for a monatomic gas. This index plays a crucial role in determining how temperature changes during volume changes without heat exchange.

The concept helps us understand basic thermodynamic properties and behaviors of gases through theoretical explorations.

Diatomic Ideal Gas

Diatomic ideal gases consist of molecules that have two atoms. Common examples include oxygen (\(O_2\)) and nitrogen (\(N_2\)). These gases are more complex than monatomic gases due to their molecular structure.
In thermodynamic calculations, diatomic gases have different properties:

• They have higher specific heat capacities because the molecules have more degrees of freedom than monatomic gases.
• The adiabatic index \( \gamma \) for diatomic gases is usually \( \frac{7}{5} \), reflecting these additional freedoms and complexity.

Understanding diatomic gases is essential for working with real-world gases, where such molecular structures are common.

Temperature Conversion

Converting temperatures from Celsius to Kelvin is essential for many scientific calculations. Kelvin is the SI unit of temperature and is often used in equations dealing with temperature, since it begins from absolute zero.
The formula for conversion is straightforward:

• \( T(K) = T(°C) + 273.15 \)
• For example, an initial temperature of 19°C converts to 292.15K.

This conversion ensures consistency and accuracy in thermodynamic calculations, enabling precise evaluations in processes such as adiabatic compression.

Adiabatic Compression

In adiabatic processes, there is no exchange of heat with the surroundings. Adiabatic compression refers to increasing a gas's pressure by reducing its volume, leading to a rise in temperature without heat transfer.
This can be described using the equation:

• \( T_{0}V_{0}^{\gamma - 1} = T_{1}V_{1}^{\gamma - 1} \), where \( T \) is temperature, \( V \) is volume, and \( \gamma \) is the adiabatic index.

For both monatomic and diatomic gases, this equation allows us to calculate the new temperature after compression, based on their specific \( \gamma \) values. This principle is fundamental in understanding thermodynamic cycles and applications like engines and refrigeration.