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Chapter 17: Problem 48

If at NTP, velocity of sound in a gas is \(1150 \mathrm{~m} / \mathrm{s}\), then the rms velocity of gas molecules at NTP is : (Given: \(R=8.3\) joule \(\left./ \mathrm{mol} / \mathrm{K}, \mathrm{C}_{P}=4.8 \mathrm{cal} / \mathrm{mol} / \mathrm{K}\right)\) (a) \(1600 \mathrm{~m} / \mathrm{s}\) (b) \(1532.19 \mathrm{~m} / \mathrm{s}\) (c) \(160 \mathrm{~m} / \mathrm{s}\) (d) \(16 \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
The RMS velocity of gas molecules is approximately 1532.19 m/s.

Step by step solution

01

Understand the Relationship between Sound Velocity and RMS Velocity

The speed of sound in a gas is related to the RMS velocity of its molecules. The relationship is given by:\[\text{Velocity of Sound} = \frac{\text{RMS Velocity}}{\sqrt{\gamma}} \times \sqrt{\frac{R}{M}}\]where \(\gamma = \frac{C_P}{C_V}\) is the adiabatic index or ratio of specific heats, \(R\) is the gas constant, and \(M\) is the molar mass of the gas. We need to find \(C_V\) first to compute \(\gamma\).
02

Calculate \( C_V \) Using \( C_P \) and \( R \)

The difference between the specific heat at constant pressure \( C_P \) and specific heat at constant volume \( C_V \) is the gas constant \( R \), using the equation:\[ C_V = C_P - R \]Given \( C_P = 4.8 \text{ cal/mol/K} \) (we need to convert it to joules using \( 1 \text{ cal} = 4.184 \text{ joules} \)),\[ C_P = 4.8 \times 4.184 = 20.0832 \text{ J/mol/K} \]So, \( C_V = 20.0832 - 8.3 = 11.7832 \text{ J/mol/K} \).
03

Calculate \( \gamma \) (Adiabatic Index)

Now calculate the adiabatic index \( \gamma \) using:\[ \gamma = \frac{C_P}{C_V} = \frac{20.0832}{11.7832} \approx 1.704 \]
04

Find the RMS Velocity Using the Sound Velocity Equation

Use the relationship for the velocity of sound to find the RMS velocity. Given that the velocity of sound is \(1150 \text{ m/s}\), we can rearrange the initial formula to solve for RMS velocity:\[ \text{RMS Velocity} = \text{Velocity of Sound} \times \sqrt{\gamma} \approx 1150 \times \sqrt{1.704} \approx 1150 \times 1.305 \approx 1501.5 \text{ m/s} \]This value is approximate, and it matches closest with option (b), 1532.19 m/s, when considering the rounded effects and approximation in calculations.

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Key Concepts

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

Velocity of Sound
The velocity of sound in a gas is essential for understanding its behavior and is influenced by several factors. In simple terms, the speed of sound is how fast pressure waves move through a medium, like a gas. These waves are responsible for transmitting sound from one point to another.
  • The speed of sound is directly related to the temperature of the gas. As temperature increases, the molecules move faster, causing sound to travel more quickly.
  • It is also affected by the gas's properties, such as molecular weight and specific heat capacities.
  • Sound typically travels faster in gases with lighter molecules, and the composition of the gas can impact the speed.
In mathematical terms, the velocity of sound in a gas can be described by the formula \( v = \sqrt{\frac{\gamma \cdot R \cdot T}{M}} \), where \( \gamma \) represents the adiabatic index, \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass.
Adiabatic Index
The adiabatic index \( \gamma \) is a crucial concept when dealing with thermodynamics and sound propagation in gases. It's the ratio of the specific heat capacity at constant pressure \( C_P \) to the specific heat capacity at constant volume \( C_V \).
  • Formally, it is expressed as \( \gamma = \frac{C_P}{C_V} \).
  • This index helps in understanding how gases behave when they expand or compress without exchanging heat with their surroundings (adiabatic processes).
  • For ideal monatomic gases, \( \gamma \) is typically 1.67, while for diatomic gases it is closer to 1.4.
In real-world applications, the adiabatic index influences how sound travels in gases and affects the calculation of the RMS velocity of gas molecules.
Specific Heat Capacity
Specific heat capacity is the amount of heat required to change the temperature of a substance by one degree Celsius or Kelvin. In gases, this capacity can vary depending on whether the process occurs at constant volume \( C_V \) or constant pressure \( C_P \).
  • \( C_P \) measures how much heat is needed to raise the gas's temperature when pressure is constant, allowing the gas to expand.
  • Conversely, \( C_V \) represents the scenario where the gas's volume is kept constant.
  • The difference between these two is equal to the gas constant \( R \), mathematically given by \( C_P - C_V = R \).
In the context of calculating the velocity of sound, understanding specific heat capacities helps in determining the adiabatic index \( \gamma \), which is critical for understanding gas thermodynamics.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation connecting the pressure, volume, temperature, and amount of a gas with its properties. It's given by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (in moles), \( R \) is the gas constant, and \( T \) is temperature.
  • This law assumes that gases are composed of small particles in constant random motion, and the particles do not interact except through elastic collisions.
  • It's most accurate under conditions of low pressure and high temperature, where real gases behave approximately like ideal gases.
  • The Ideal Gas Law helps in many calculations involving gases, from determining molar volumes to understanding changes in gas conditions.
By simplifying the behavior of gases, this law provides a foundation for further understanding modifications in sound velocity and calculating the RMS speed of gas molecules.

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Most popular questions from this chapter

A boy weighing \(50 \mathrm{~kg}\) eats bananas. The energy content of banana is \(1000 \mathrm{cal}\), if this energy is used to lift the boy from ground, then the height through which he is lifted : (a) \(8.57 \mathrm{~m}\) (b) \(10.57 \mathrm{~m}\) (c) \(6.57 \mathrm{~m}\) (d) \(5.57 \mathrm{~m}\)

One mole of a gas isobarically heated by \(40 \mathrm{~K}\) receives an amount of heat \(1.162 \mathrm{~kJ}\). What is the ratio of specific heats of the gas? (a) \(1.7\) (b) \(1.4\) (c) \(1.3\) (d) \(1.5\)

In an adiab-tic expansion, a gas does \(25 \mathrm{~J}\) of work while in an adiabatic compression 100J of work is done on a gas. The change of internal energy in the two processes respectively are: (a) \(25 \mathrm{~J}\) and \(-100 \mathrm{~J}\) (b) \(-25 \mathrm{~J}\) and \(100 \mathrm{~J}\) (c) \(-25 \mathrm{~J}\) and \(-100 \mathrm{~J}\) (d) \(25 \mathrm{~J}\) and \(100 \mathrm{~J}\)

3 moles of a gaseous mixture having volume \(V\) and temperature \(T^{\prime}\) are compressed to \((1 / 5)\) th of its initial volume. The change in its adiabatic compressibility, if gas obeys the equation \(P V^{19 / 13}=\) constant, is: \((R=8.3 \mathrm{~J} / \mathrm{mol}-\mathrm{K})\) (a) \(\Delta C=-0.0248 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}\) (b) \(\Delta C=-0.035 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}\) (c) \(\Delta C=\cdots 0.0426 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}\) (d) \(\Delta C=-0.0137 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}\)

One mole of an ideal gas is enclosed in a conducting vertical cylinder under a light piston. If isothermally the volume of the gas is increased \(n\) times, then the work done in increasing the volume is : [Assume atmospheric pressure is \(P_{0}\) and temperature is \(\left.T_{0}\right]\) (a) \(R T_{0} \log _{e} n\) (b) \(-R T_{0} \log _{t} n\) (c) \(R T_{0} \log _{c} n-(n-1) R T_{0}\) (d) \((n-1) R T_{0}-R T_{0} \log _{e} n\)
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