A Piston Contains 0.232 Mol Of A Diatomic Gas. The Piston Is Compressed While 57.5 J Of Heat Are Removed. The Temperature Changes By $68.0^{\circ} C$.How Much Work Is Done By The Gas? W = [ ? ] J W = [?] \, \text{J} W = [ ?] J

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Introduction

In thermodynamics, the work done by a gas is a crucial concept that helps us understand the energy transfer between the system and its surroundings. The work done by a gas can be calculated using the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In this problem, we are given a piston containing 0.232 mol of a diatomic gas, and we are asked to find the work done by the gas when it is compressed while 57.5 J of heat are removed, and the temperature changes by $68.0^{\circ} C$.

The First Law of Thermodynamics

The first law of thermodynamics is a fundamental principle in thermodynamics that relates the change in internal energy of a system to the heat added to the system and the work done by the system. It is expressed mathematically as:

$$\Delta U = Q - W$$

where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system.

The Internal Energy of a Diatomic Gas

The internal energy of a diatomic gas is a function of its temperature. For a diatomic gas, the internal energy is given by:

$$U = \frac{5}{2}nRT$$

where $n$ is the number of moles of gas, $R$ is the gas constant, and $T$ is the temperature in Kelvin.

The Work Done by the Gas

The work done by the gas can be calculated using the first law of thermodynamics. We are given that the heat removed from the gas is 57.5 J, and the temperature change is $68.0^{\circ} C$. We can use the first law of thermodynamics to calculate the work done by the gas.

Calculating the Work Done by the Gas

To calculate the work done by the gas, we need to first calculate the change in internal energy of the gas. We can do this by using the equation for the internal energy of a diatomic gas:

$$\Delta U = \frac{5}{2}nR\Delta T$$

where $\Delta T$ is the change in temperature.

We are given that the number of moles of gas is 0.232 mol, and the temperature change is $68.0^{\circ} C$. We can plug these values into the equation to get:

$$\Delta U = \frac{5}{2}(0.232 \, \text{mol})(8.314 \, \text{J/mol}\cdot\text{K})(68.0 \, \text{K})$$

$$\Delta U = 143.1 \, \text{J}$$

Now that we have calculated the change in internal energy of the gas, we can use the first law of thermodynamics to calculate the work done by the gas:

$$W = Q - \Delta U$$

We are given that the heat removed from the gas is 57.5 J, so we can plug this value into the equation to get:

$$W = 57.5 \, \text{J} - 143.1 \, \text{J}$$

$$W = -85.6 \, \text{J}$$

However, the work done by the gas is a positive quantity, so we take the absolute value of the result:

$$W = 85.6 \, \text{J}$$

Conclusion

In this problem, we were given a piston containing 0.232 mol of a diatomic gas, and we were asked to find the work done by the gas when it is compressed while 57.5 J of heat are removed, and the temperature changes by $68.0^{\circ} C$. We used the first law of thermodynamics to calculate the work done by the gas, and we found that the work done by the gas is 85.6 J.

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] Zemansky, M. W., & Dittman, R. H. (2013). Heat and thermodynamics: An intermediate textbook. McGraw-Hill Education.

Discussion

The work done by a gas is an important concept in thermodynamics that helps us understand the energy transfer between the system and its surroundings. In this problem, we used the first law of thermodynamics to calculate the work done by the gas, and we found that the work done by the gas is 85.6 J. This result is consistent with the expected behavior of a diatomic gas under compression.

Additional Information

• The work done by a gas can be calculated using the first law of thermodynamics.
• The internal energy of a diatomic gas is a function of its temperature.
• The work done by a gas is a positive quantity.
• The first law of thermodynamics is a fundamental principle in thermodynamics that relates the change in internal energy of a system to the heat added to the system and the work done by the system.
  

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Q&A

Q: What is the first law of thermodynamics?

A: The first law of thermodynamics is a fundamental principle in thermodynamics that relates the change in internal energy of a system to the heat added to the system and the work done by the system. It is expressed mathematically as:

$$\Delta U = Q - W$$

where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system.

Q: What is the internal energy of a diatomic gas?

A: The internal energy of a diatomic gas is a function of its temperature. For a diatomic gas, the internal energy is given by:

$$U = \frac{5}{2}nRT$$

where $n$ is the number of moles of gas, $R$ is the gas constant, and $T$ is the temperature in Kelvin.

Q: How is the work done by a gas calculated?

A: The work done by a gas can be calculated using the first law of thermodynamics. We can use the equation:

$$W = Q - \Delta U$$

where $Q$ is the heat added to the system, and $\Delta U$ is the change in internal energy of the system.

Q: What is the significance of the work done by a gas?

A: The work done by a gas is an important concept in thermodynamics that helps us understand the energy transfer between the system and its surroundings. It is a measure of the energy transferred from the system to the surroundings due to a change in the system's state.

Q: Can the work done by a gas be negative?

A: No, the work done by a gas cannot be negative. The work done by a gas is a positive quantity, and it represents the energy transferred from the system to the surroundings.

Q: What is the relationship between the work done by a gas and the heat added to the system?

A: The work done by a gas is related to the heat added to the system through the first law of thermodynamics. The work done by a gas is equal to the heat added to the system minus the change in internal energy of the system.

Q: Can the work done by a gas be zero?

A: No, the work done by a gas cannot be zero. The work done by a gas is a positive quantity, and it represents the energy transferred from the system to the surroundings.

Q: What is the significance of the temperature change in calculating the work done by a gas?

A: The temperature change is an important factor in calculating the work done by a gas. The temperature change affects the internal energy of the gas, which in turn affects the work done by the gas.

Q: Can the work done by a gas be calculated using other methods?

A: Yes, the work done by a gas can be calculated using other methods, such as the ideal gas equation and the equation of state for a real gas. However, the first law of thermodynamics is a fundamental principle that provides a general framework for calculating the work done by a gas.

Additional Information

• The work done by a gas is an important concept in thermodynamics that helps us understand the energy transfer between the system and its surroundings.
• The first law of thermodynamics is a fundamental principle that relates the change in internal energy of a system to the heat added to the system and the work done by the system.
• The internal energy of a diatomic gas is a function of its temperature.
• The work done by a gas is a positive quantity.
• The temperature change is an important factor in calculating the work done by a gas.

Discussion

The work done by a gas is an important concept in thermodynamics that helps us understand the energy transfer between the system and its surroundings. In this article, we have discussed the first law of thermodynamics, the internal energy of a diatomic gas, and the calculation of the work done by a gas. We have also answered some common questions related to the work done by a gas.