A Car Engine Burns Gas At 612 K 612 \, K 612 K And Dumps Heat Into The Air At 245 K 245 \, K 245 K . The Car Runs At Carnot Efficiency. How Much Input Heat Would The Car Require To Do 5 , 000 J 5,000 \, J 5 , 000 J Of Work? Q = [ ? ] J Q = \, [?] \, J Q = [ ?] J

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Introduction

The Carnot engine is a theoretical engine that operates on the Carnot cycle, which is an idealized thermodynamic cycle. It is used to calculate the maximum possible efficiency of a heat engine. The Carnot efficiency is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

Carnot Efficiency

The Carnot efficiency is a measure of the maximum possible efficiency of a heat engine. It is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

Work Done by the Car Engine

The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

Heat Input

To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Calculation

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

Conclusion

The car engine requires $8,326 \, J$ of input heat to do $5,000 \, J$ of work.

References

• Carnot, S. (1824). Reflections on the Motive Power of Fire.
• Feynman, R. P. (1963). The Feynman Lectures on Physics.

Discussion

The Carnot engine is a theoretical engine that operates on the Carnot cycle, which is an idealized thermodynamic cycle. It is used to calculate the maximum possible efficiency of a heat engine. The Carnot efficiency is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

The car engine requires $8,326 \, J$ of input heat to do $5,000 \, J$ of work.

Additional Discussion

The Carnot engine is a theoretical engine that operates on the Carnot cycle, which is an idealized thermodynamic cycle. It is used to calculate the maximum possible efficiency of a heat engine. The Carnot efficiency is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

The car engine requires $8,326 \, J$ of input heat to do $5,000 \, J$ of work.

Real-World Applications

The Carnot engine is a theoretical engine that operates on the Carnot cycle, which is an idealized thermodynamic cycle. It is used to calculate the maximum possible efficiency of a heat engine. The Carnot efficiency is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

The car engine requires $8,326 \, J$ of input heat to do $5,000 \, J$ of work.

Conclusion

The Carnot engine is a theoretical engine that operates on the Carnot cycle, which is an idealized thermodynamic cycle. It is used to calculate the maximum possible efficiency of a heat engine. The Carnot efficiency is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

The car engine requires $8,326 \, J$ of input heat to do $5,000 \, J$ of work.

Final Answer

The final answer is: $\boxed{8326}$

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Q&A

Q: What is the Carnot efficiency of the car engine?

A: The Carnot efficiency of the car engine is given by the equation:

$$\eta = 1 - \frac{T_c}{T_h}$$

where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.

In this problem, the car engine burns gas at $612 \, K$ and dumps heat into the air at $245 \, K$. The Carnot efficiency of the car engine is:

$$\eta = 1 - \frac{245 \, K}{612 \, K} = 0.599$$

Q: What is the work done by the car engine?

A: The work done by the car engine is given by the equation:

$$W = \eta Q_h$$

where $W$ is the work done, $\eta$ is the Carnot efficiency, and $Q_h$ is the heat input.

We are given that the car engine does $5,000 \, J$ of work. We can use the equation above to find the heat input:

$$5,000 \, J = 0.599 Q_h$$

Q: How much heat input is required to do $5,000 \, J$ of work?

A: To find the heat input, we can rearrange the equation above to solve for $Q_h$:

$$Q_h = \frac{5,000 \, J}{0.599}$$

Now we can calculate the heat input:

$$Q_h = \frac{5,000 \, J}{0.599} = 8,326 \, J$$

Q: What is the significance of the Carnot efficiency in this problem?

A: The Carnot efficiency is a measure of the maximum possible efficiency of a heat engine. In this problem, the car engine runs at Carnot efficiency, which means that it is operating at its maximum possible efficiency.

Q: How does the Carnot efficiency affect the heat input required to do $5,000 \, J$ of work?

A: The Carnot efficiency affects the heat input required to do $5,000 \, J$ of work by determining the amount of heat that is converted into work. In this problem, the Carnot efficiency is 0.599, which means that 59.9% of the heat input is converted into work.

Q: What is the relationship between the heat input and the work done by the car engine?

A: The heat input and the work done by the car engine are related by the equation:

$$W = \eta Q_h$$

This equation shows that the work done by the car engine is directly proportional to the heat input and the Carnot efficiency.

Q: How can the Carnot efficiency be improved in a real-world engine?

A: The Carnot efficiency can be improved in a real-world engine by increasing the temperature of the hot reservoir or decreasing the temperature of the cold reservoir. This can be achieved by using more efficient heat transfer methods or by using materials with higher thermal conductivity.

Q: What are some real-world applications of the Carnot engine?

A: The Carnot engine has several real-world applications, including:

• Power generation: The Carnot engine can be used to generate electricity by converting heat into work.
• Refrigeration: The Carnot engine can be used to refrigerate materials by transferring heat from a cold reservoir to a hot reservoir.
• Heat pumps: The Carnot engine can be used to heat buildings by transferring heat from a cold reservoir to a hot reservoir.

Q: What are some limitations of the Carnot engine?

A: The Carnot engine has several limitations, including:

• Efficiency: The Carnot engine is limited by its efficiency, which is determined by the Carnot efficiency.
• Temperature: The Carnot engine is limited by the temperature of the hot and cold reservoirs.
• Materials: The Carnot engine is limited by the materials used to construct it.

Q: What are some future directions for the Carnot engine?

A: Some future directions for the Carnot engine include:

• Improving efficiency: Researchers are working to improve the efficiency of the Carnot engine by developing new materials and heat transfer methods.
• Increasing temperature: Researchers are working to increase the temperature of the hot reservoir to improve the efficiency of the Carnot engine.
• Developing new applications: Researchers are working to develop new applications for the Carnot engine, such as power generation and refrigeration.