When The Temperature Of A Gas Is Increased By 300K, The Velocity Of Sound In The Gas Becomes 2 Times Its Initial Velocity. What Is The Initial Temperature Of The Gas?1) 300°C2) 27°C3) 270°C4) 30°C

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Understanding the Relationship Between Temperature and Velocity of Sound

When the temperature of a gas is increased, the velocity of sound in the gas also changes. This phenomenon is a fundamental concept in physics, and understanding it requires a grasp of the underlying principles. In this article, we will delve into the relationship between temperature and the velocity of sound in a gas, and use this knowledge to solve a problem involving a specific temperature increase.

The Speed of Sound in a Gas

The speed of sound in a gas is determined by several factors, including the temperature, pressure, and composition of the gas. However, for the purpose of this discussion, we will focus on the relationship between temperature and the speed of sound. The speed of sound in a gas is given by the equation:

c = √(γRT/M)

where c is the speed of sound, γ is the adiabatic index, R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.

The Effect of Temperature on the Speed of Sound

As the temperature of a gas increases, the speed of sound also increases. This is because the molecules of the gas gain kinetic energy and move faster, allowing the sound wave to propagate more quickly. The relationship between temperature and the speed of sound is given by the equation:

c ∝ √T

This equation shows that the speed of sound is directly proportional to the square root of the temperature.

The Problem

The problem states that when the temperature of a gas is increased by 300K, the velocity of sound in the gas becomes 2 times its initial velocity. We are asked to find the initial temperature of the gas.

Let's denote the initial temperature as T1 and the final temperature as T2. We know that the speed of sound is directly proportional to the square root of the temperature, so we can write:

c2/c1 = √(T2/T1)

We are given that the final temperature is 300K higher than the initial temperature, so we can write:

T2 = T1 + 300

Substituting this expression into the previous equation, we get:

c2/c1 = √((T1 + 300)/T1)

We are also given that the final velocity is 2 times the initial velocity, so we can write:

c2 = 2c1

Substituting this expression into the previous equation, we get:

2 = √((T1 + 300)/T1)

Squaring both sides of the equation, we get:

4 = (T1 + 300)/T1

Multiplying both sides of the equation by T1, we get:

4T1 = T1 + 300

Subtracting T1 from both sides of the equation, we get:

3T1 = 300

Dividing both sides of the equation by 3, we get:

T1 = 100

Therefore, the initial temperature of the gas is 100K.

Conclusion

In this article, we have explored the relationship between temperature and the velocity of sound in a gas. We have used this knowledge to solve a problem involving a specific temperature increase, and have found the initial temperature of the gas to be 100K. This problem demonstrates the importance of understanding the underlying principles of physics, and how they can be applied to real-world situations.

Additional Information

  • The adiabatic index (γ) is a dimensionless quantity that depends on the composition of the gas. For a monatomic gas, γ = 5/3, while for a diatomic gas, γ = 7/5.
  • The gas constant (R) is a constant that depends on the units of temperature and pressure. For example, R = 8.314 J/mol·K in the International System of Units (SI).
  • The molar mass (M) of a gas is the mass of one mole of the gas. For example, the molar mass of air is approximately 28.97 g/mol.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
  • [3] Young, H. D., & Freedman, R. A. (2015). University physics. Addison-Wesley.
    Frequently Asked Questions: Temperature and Velocity of Sound

In our previous article, we explored the relationship between temperature and the velocity of sound in a gas. We also solved a problem involving a specific temperature increase and found the initial temperature of the gas to be 100K. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the relationship between temperature and the speed of sound in a gas?

A: The speed of sound in a gas is directly proportional to the square root of the temperature. This means that as the temperature of a gas increases, the speed of sound also increases.

Q: How does the adiabatic index (γ) affect the speed of sound in a gas?

A: The adiabatic index (γ) is a dimensionless quantity that depends on the composition of the gas. For a monatomic gas, γ = 5/3, while for a diatomic gas, γ = 7/5. The adiabatic index affects the speed of sound in a gas by changing the relationship between temperature and speed.

Q: What is the gas constant (R) and how does it affect the speed of sound in a gas?

A: The gas constant (R) is a constant that depends on the units of temperature and pressure. For example, R = 8.314 J/mol·K in the International System of Units (SI). The gas constant affects the speed of sound in a gas by changing the relationship between temperature and speed.

Q: How does the molar mass (M) of a gas affect the speed of sound in a gas?

A: The molar mass (M) of a gas is the mass of one mole of the gas. For example, the molar mass of air is approximately 28.97 g/mol. The molar mass affects the speed of sound in a gas by changing the relationship between temperature and speed.

Q: Can the speed of sound in a gas be affected by other factors besides temperature?

A: Yes, the speed of sound in a gas can be affected by other factors besides temperature. These factors include pressure, composition, and the presence of impurities.

Q: How does the speed of sound in a gas change with altitude?

A: The speed of sound in a gas changes with altitude due to changes in temperature and pressure. At higher altitudes, the temperature and pressure are lower, which results in a lower speed of sound.

Q: Can the speed of sound in a gas be affected by the presence of a magnetic field?

A: No, the speed of sound in a gas is not affected by the presence of a magnetic field. The speed of sound in a gas is determined by the properties of the gas itself, such as its temperature, pressure, and composition.

Q: How does the speed of sound in a gas compare to the speed of sound in a solid?

A: The speed of sound in a gas is typically much lower than the speed of sound in a solid. This is because the molecules in a gas are free to move and collide with each other, whereas the molecules in a solid are fixed in place and cannot move.

Q: Can the speed of sound in a gas be used to measure temperature?

A: Yes, the speed of sound in a gas can be used to measure temperature. By measuring the speed of sound in a gas and using the relationship between temperature and speed, it is possible to determine the temperature of the gas.

Conclusion

In this article, we have answered some frequently asked questions related to the relationship between temperature and the velocity of sound in a gas. We have discussed the adiabatic index, gas constant, molar mass, and other factors that affect the speed of sound in a gas. We have also compared the speed of sound in a gas to the speed of sound in a solid and discussed the use of the speed of sound in a gas to measure temperature.

Additional Information

  • The speed of sound in a gas can be affected by the presence of impurities, such as dust or water vapor.
  • The speed of sound in a gas can be measured using a variety of techniques, including the Doppler effect and interferometry.
  • The speed of sound in a gas is an important parameter in many fields, including physics, engineering, and medicine.

References

  • [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
  • [2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.
  • [3] Young, H. D., & Freedman, R. A. (2015). University physics. Addison-Wesley.