Question

If the mass of 1 mole of air is \(29 \times 10^{-3} \mathrm{~kg}\), then the speed of sound in it at STP is \((\gamma=7 / 5) .\left\\{\mathrm{T}=273 \mathrm{~K}, \mathrm{P}=1.01 \times 10^{5} \mathrm{~Pa}\right\\}\) (A) \(270 \mathrm{~m} / \mathrm{s}\) (B) \(290 \mathrm{~m} / \mathrm{s}\) (C) \(330 \mathrm{~m} / \mathrm{s}\) (D) \(350 \mathrm{~m} / \mathrm{s}\)

Short answer

The speed of sound in air at STP is approximately \(330 \mathrm{~m} / \mathrm{s}\). The correct answer is (C).

Step-by-step solution

Since the mass of 1 mole of air is given as \(29 \times 10^{-3}\mathrm{kg}\), we can calculate the gas constant \(R\) using the given pressure and temperature. The formula for the gas constant is: \( PV=nRT\) For 1 mole of gas, n=1, so \( R = \frac{PV}{T}\) #Step 2: Plug in the values and calculate R#

We are given the values of P and T. So, let's plug them into the formula and solve for R: \( R = \frac{1.01\times10^5Pa}{(1)(273K)}\) Calculating R gives us: \(R \approx 370.3 \mathrm{J} / \mathrm{kg}\cdot\mathrm{K}\) #Step 3: Calculate the speed of sound using the formula#

Now that we have the value of R, we can use the formula for the speed of sound in an ideal gas: \( v = \sqrt{\frac{\gamma RT}{M}}\) We know the values of γ, R, T, and M. So we can plug them into the formula: \( v = \sqrt{\frac{(7/5)(370.3)(273)}{29\times10^{-3}}}\) #Step 4: Solve for the speed of sound v#

Now we can solve for v: \(v \approx 331 \mathrm{~m} / \mathrm{s}\) Comparing this to the available options, the speed of sound in air at STP is closest to 330 m/s. The correct answer is (C) \(330 \mathrm{~m} / \mathrm{s}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial equation in thermodynamics that relates the pressure, volume, temperature, and amount of gas. It is represented by the formula \(PV = nRT\), where:

• \(P\) stands for pressure.
• \(V\) represents volume.
• \(n\) is the number of moles.
• \(R\) is the ideal gas constant, a universal constant that can be calculated given certain conditions.
• \(T\) denotes the temperature in Kelvin.

For the speed of sound calculation, knowing the Ideal Gas Law allows us to derive the gas constant \(R\) when mass, pressure, and temperature are known. This law simplifies understanding how gases behave under different conditions and is fundamental in calculating properties such as speed of sound in gases.

Molar Mass

Molar Mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). In our problem, we deal with the molar mass of air, which is given as \(29 \times 10^{-3} \mathrm{~kg}\).

This value enables us to determine other properties of air, like how it behaves as a gas under typical conditions. Knowing the molar mass is necessary for using other thermodynamic equations, such as those calculating speed of sound, where it is included in the denominator to compute density from mass and volume.

For physical properties that depend on the mass and amount of matter, like gas behavior or reactions involving gases, the molar mass links the microscopic properties of molecules to macroscopic behaviors.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, or STP, is a set of conditions used to describe measurements in the field of chemistry. It provides a consistent basis for the comparison of properties of gases.

STP is defined by a temperature of 273 Kelvin and a pressure of \(1.01 \times 10^5\) Pascals. Using STP simplifies calculations and comparisons between different gases and conditions.

In the context of our speed of sound calculation, knowing the properties of air at STP allows us to use consistent values for temperature and pressure, making the calculation straightforward and standardized across practical applications.

Adiabatic Index (Gamma)

The Adiabatic Index, also known as Gamma (\(\gamma\)), is an important factor in thermodynamics when dealing with processes where heat energy changes but the substance does not exchange heat with its environment.

Gamma is the ratio of specific heats at constant pressure (\(C_p\)) to that at constant volume (\(C_v\)). For air, this ratio is typically \(\frac{7}{5}\) or 1.4.

This value reflects how a gas will expand or compress under adiabatic processes. In speed of sound calculations, gamma directly impacts the equation \(v = \sqrt{\frac{\gamma RT}{M}}\), demonstrating how it affects sound propagation in gases. It is essential for predicting and understanding how changes in pressure and volume without heat exchange affect gases.