91Ó°ÊÓ

A long, closed cylindrical tank contains a diatomic gas that is maintained at a uniform temperature that can be varied. When you measure the speed of sound \(v\) in the gas as a function of the temperature \(T\) of the gas, you obtain these results: $$ \begin{array}{l|cccccc} \boldsymbol{T}\left({ }^{\circ} \mathbf{C}\right) & -20.0 & 0.0 & 20.0 & 40.0 & 60.0 & 80.0 \\ \hline \boldsymbol{v}(\mathbf{m} / \mathrm{s}) & 324 & 337 & 349 & 361 & 372 & 383 \end{array} $$ (a) Explain how you can plot these results so that the graph will be well fit by a straight line. Construct this graph and verify that the plotted points do lie close to a straight line. (b) Because the gas is diatomic, \(\gamma=1.40 .\) Use the slope of the line in part (a) to calculate \(M\) the molar mass of the gas. Express \(M\) in grams/imole. What type of gas is in the tank?

Step by step solution

01

Plotting the graph

The data provided corresponds to the velocity of sound in the gas \(v\) and the gas temperature \(T\) in degrees Celsius. But, the formula of the velocity of sound in a gas is \(v = \sqrt{\gamma R T/M}\), where \(\gamma\) is the heat capacity ratio, \(R\) is the universal gas constant, \(T\) is the temperature and \(M\) is the molar mass. To get a straight line, the plotting should be done against \(\sqrt{T}\) and not \(T\). So first convert the temperature \(T\) from Celsius to Kelvin by adding 273.15 to each of the given temperatures. Then, take the square root of each of the temperature values.

02

Verify that the points lie close to a straight line

Next, plot the values of \(v\) on the y-axis and \(\sqrt{T}\) on the x-axis. Visually inspect the graph to verify if the points align closely to a straight line. The alignment is an indication that the velocity of sound in the gas is indeed governed by the relation \(v = \sqrt{\gamma R T/M}\).

03

Calculate Molar mass

From the line of best fit on the plot, obtain the slope. The slope of the line equals to \(\sqrt{\gamma R/M}\). To calculate the molar mass \(M\), rearrange the formula to become \(M = \gamma R / slope^2\). Plug in \( \gamma = 1.40\), \( R = 8.31 J/(mol.K)\) and the value of the slope obtained from the graph to calculate the molar mass.

04

Identify the gas

Molar mass is a unique property that can be used to identify substances. Lookup tables or use internet sources to match the molar mass calculated above with known gases. The gas with the closest matching molar mass is the gas in the tank.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diatomic Gas

Diatomic gases are molecules composed of two atoms. Common examples include oxygen \\((O_2)\), nitrogen \\((N_2)\), and hydrogen \\((H_2)\). In the context of physics and chemistry, diatomic gases play an important role. These gases have unique properties, such as specific heat capacities and molecular structures.
The presence of two atoms in a diatomic molecule leads to

• More complex interactions than monoatomic gases.
• A higher heat capacity, as energy can be stored in rotational and vibrational modes in addition to translational motion.

For sound traveling through gases, such as those in the cylindrical tank from our exercise, the way these diatomic molecules interact has a significant effect. The speed of sound is faster in diatomic gases than in monoatomic gases when temperature and pressure conditions are similar. This has to do with the energy distribution and the gas's molecular mass.

Molar Mass Calculation

Molar mass is crucial in determining the identity of a gas. It represents the mass of one mole of a substance typically expressed in grams per mole \\((g/mol)\). In gases, knowing the molar mass helps predict and understand gas behavior under various conditions using equations like the ideal gas law.
To calculate molar mass from the given speed of sound data, you follow these steps:

• Use the formula: \( v = \sqrt{\frac{\gamma R T}{M}} \), where \(v\) is the sound velocity, \(\gamma\) is the heat capacity ratio, \(R\) is the universal gas constant, \(T\) is temperature, and \(M\) is molar mass.
• Rearrange to solve for \(M\): \( M = \frac{\gamma R}{slope^2} \).
• By knowing \(\gamma = 1.40\) for diatomic gases and \(R = 8.31 \) \(J/(mol \cdot K)\), you can determine \(M\).

Through this process, we can identify what gas is inside the tank based on the calculated molar mass.

Heat Capacity Ratio

The heat capacity ratio, \(\gamma\), is a dimensionless number important in thermodynamics and fluid dynamics. It is the ratio of a gas's heat capacity at constant pressure \((C_p)\) to its heat capacity at constant volume \((C_v)\). For diatomic gases, \(\gamma\) is generally about 1.40. This value can slightly vary depending on temperature and other conditions.
This ratio is vital because:

• It affects the sound speed in the gas. Higher \(\gamma\) usually results in higher sound speed.
• It influences how gases react to compression and expansion, which is important in engines and heat exchangers.

Knowing \(\gamma\) helps in solving thermodynamic problems by linking pressure, volume, and temperature changes.

Temperature Conversion

Temperature conversion is a necessary skill when working with various scientific equations, particularly when moving between the Celsius and Kelvin scales. Sound speed in gases, for instance, depends on absolute temperature measured in Kelvin.
To convert Celsius to Kelvin:

• Add 273.15 to the Celsius value.
• Kelvin is the standard unit used in thermodynamic calculations due to its absolute scale.

In the exercise, temperatures are initially given in Celsius, so they are converted to Kelvin before calculating sound speed. This ensures that temperature values are compatible with the universal gas constant \(R\) in equations.

Universal Gas Constant

The universal gas constant, \(R\), is a fundamental constant in thermodynamics and physical chemistry. It appears in equations such as the ideal gas law and relates energy scales in thermodynamic processes. Its value is \(8.31 \) \(J/(mol \cdot K)\).
This constant connects:

• Pressure, volume, and temperature with the amount of substance.
• Appears in the equation of speed of sound: \( v = \sqrt{\frac{\gamma R T}{M}} \).

With \(R\) being a key component in calculations involving gas laws, it allows us to understand and predict gas behavior and interactions. It remains a core part of evaluating gas parameters like speed of sound, molar volume, and internal energy changes.