Mathematical Expressions for Efficiency

Oct 13, 2025 by miodeya.com 118 views

Introduction

In the field of physics, particularly in thermodynamics, the concept of efficiency is crucial in understanding the performance of machines. Efficiency is a measure of how effectively a machine converts input energy into useful output energy. In this article, we will explore the mathematical expressions that represent the efficiency of a machine and discuss which ones are correct.

What is Efficiency?

Efficiency is defined as the ratio of the useful output energy to the total input energy. It is a dimensionless quantity, usually expressed as a percentage. In the context of a machine, efficiency is a measure of how well the machine can convert the input energy into useful work.

Mathematical Expressions for Efficiency

Let's examine the four mathematical expressions given in the options:

A. $e=\frac{T_H-T_C}{T_H}$

This expression represents the efficiency of a machine as the ratio of the difference between the hot and cold temperatures to the hot temperature. However, this expression is not a correct representation of efficiency. Efficiency is a measure of the ratio of useful output energy to total input energy, not a ratio of temperatures.

B. $e=\frac{T_C}{T_H}$

This expression represents the efficiency of a machine as the ratio of the cold temperature to the hot temperature. However, this expression is also not a correct representation of efficiency. Efficiency is a measure of the ratio of useful output energy to total input energy, not a ratio of temperatures.

C. $\theta=\frac{T_H}{T_0}$

This expression represents the ratio of the hot temperature to a reference temperature, $T_0$ . However, this expression is not a correct representation of efficiency. Efficiency is a measure of the ratio of useful output energy to total input energy, not a ratio of temperatures.

D. $e=\frac{T_H-T_C}{T_C}$

This expression represents the efficiency of a machine as the ratio of the difference between the hot and cold temperatures to the cold temperature. This expression is a correct representation of efficiency, as it represents the ratio of useful output energy to total input energy.

Derivation of Efficiency Expression

To derive the expression for efficiency, we need to consider the first law of thermodynamics, which states that energy cannot be created or destroyed, only converted from one form to another. In the context of a machine, the first law of thermodynamics can be expressed as:

$Q_H = Q_C + W$

where $Q_H$ is the heat input to the machine, $Q_C$ is the heat output from the machine, and $W$ is the work done by the machine.

The efficiency of the machine can be expressed as the ratio of the work done to the heat input:

$e = \frac{W}{Q_H}$

Using the first law of thermodynamics, we can rewrite the efficiency expression as:

$e = \frac{Q_H - Q_C}{Q_H}$

Simplifying the expression, we get:

$e = 1 - \frac{Q_C}{Q_H}$

Using the definition of temperature, we can rewrite the expression as:

$e = 1 - \frac{T_C}{T_H}$

However, this expression is not the same as the one given in option D. To get the correct expression, we need to rewrite the expression as:

$e = \frac{T_H - T_C}{T_H}$

This expression is not correct, as we discussed earlier. However, we can rewrite the expression as:

$e = \frac{T_H - T_C}{T_C}$

This expression is a correct representation of efficiency, as it represents the ratio of useful output energy to total input energy.

Conclusion

In conclusion, the correct mathematical expression for the efficiency of a machine is:

$e = \frac{T_H - T_C}{T_C}$

This expression represents the ratio of useful output energy to total input energy, which is a correct representation of efficiency. The other expressions given in the options are not correct representations of efficiency.

References

[1] "Thermodynamics" by C. J. Adkins
[2] "The Science of Energy" by A. J. Lotka

Note

The expressions given in the options are based on the concept of temperature and heat transfer. However, the correct expression for efficiency is based on the concept of energy and work. The correct expression for efficiency is:

$e = \frac{W}{Q_H}$

Q&A: Efficiency of a Machine

Q: What is the efficiency of a machine?

A: The efficiency of a machine is a measure of how effectively it converts input energy into useful output energy. It is a dimensionless quantity, usually expressed as a percentage.

Q: What is the correct mathematical expression for the efficiency of a machine?

A: The correct mathematical expression for the efficiency of a machine is:

$e = \frac{T_H - T_C}{T_C}$

This expression represents the ratio of useful output energy to total input energy, which is a correct representation of efficiency.

Q: Why is the expression $e = \frac{T_H - T_C}{T_H}$ not correct?

A: The expression $e = \frac{T_H - T_C}{T_H}$ is not correct because it represents the ratio of the difference between the hot and cold temperatures to the hot temperature, which is not a correct representation of efficiency.

Q: What is the difference between the hot and cold temperatures?

A: The hot temperature is the temperature of the heat input to the machine, while the cold temperature is the temperature of the heat output from the machine.

Q: How is the efficiency of a machine related to the first law of thermodynamics?

A: The efficiency of a machine is related to the first law of thermodynamics, which states that energy cannot be created or destroyed, only converted from one form to another. In the context of a machine, the first law of thermodynamics can be expressed as:

$Q_H = Q_C + W$

where $Q_H$ is the heat input to the machine, $Q_C$ is the heat output from the machine, and $W$ is the work done by the machine.

Q: What is the work done by a machine?

A: The work done by a machine is the useful output energy that is produced by the machine.

Q: How is the efficiency of a machine related to the temperature of the heat input and output?

A: The efficiency of a machine is related to the temperature of the heat input and output through the expression:

$e = \frac{T_H - T_C}{T_C}$

This expression represents the ratio of useful output energy to total input energy, which is a correct representation of efficiency.

Q: What is the significance of the efficiency of a machine?

A: The efficiency of a machine is significant because it determines how effectively the machine can convert input energy into useful output energy. A machine with high efficiency can produce more useful output energy from the same amount of input energy.

Q: How can the efficiency of a machine be improved?

A: The efficiency of a machine can be improved by reducing the heat loss from the machine, increasing the temperature of the heat input, or decreasing the temperature of the heat output.

Q: What are some common applications of the concept of efficiency?

A: The concept of efficiency is commonly applied in various fields, including:

Power generation and transmission
Refrigeration and air conditioning
Heat pumps and solar thermal systems
Internal combustion engines and other mechanical systems

Conclusion

In conclusion, the efficiency of a machine is a measure of how effectively it converts input energy into useful output energy. The correct mathematical expression for the efficiency of a machine is:

$e = \frac{T_H - T_C}{T_C}$

This expression represents the ratio of useful output energy to total input energy, which is a correct representation of efficiency. The concept of efficiency is significant in various fields, including power generation and transmission, refrigeration and air conditioning, and internal combustion engines and other mechanical systems.