Calculating the Pressure at a Given Temperature

Oct 13, 2025 by miodeya.com 70 views

Introduction

The pressure inside a submarine cabin is a critical factor that affects the comfort and safety of its occupants. As the cabin temperature changes, the pressure inside the cabin also changes due to the ideal gas law. In this article, we will explore the relationship between temperature and pressure in a submarine cabin and provide a numerical setup for calculating the pressure at a given temperature.

The Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that describes the behavior of ideal gases. It states that the product of the pressure (P) and volume (V) of a gas is equal to the product of the number of moles (n) of the gas and the gas constant (R) multiplied by the temperature (T) in Kelvin:

P × V = n × R × T

Given Information

The enclosed cabin of a submarine has:

A volume of $2.4 \times 10^3$ liters
An initial temperature of 293 K
A pressure of 1 atm (101,325 Pa)

Calculating the Pressure at a Given Temperature

To calculate the pressure at a given temperature, we can rearrange the ideal gas law to solve for pressure:

P = n × R × T / V

We are given the initial temperature (T = 293 K), volume (V = $2.4 \times 10^3$ liters), and pressure (P = 1 atm). We can use these values to calculate the number of moles (n) of the gas:

n = P × V / (R × T)

Using the gas constant (R = 8.3145 J/mol·K), we can calculate the number of moles:

n = 101,325 Pa × $2.4 \times 10^3$ L / (8.3145 J/mol·K × 293 K) ≈ 9.32 mol

Now that we have the number of moles, we can calculate the pressure at a given temperature (T = 293 K) using the ideal gas law:

P = n × R × T / V

Substituting the values, we get:

P = 9.32 mol × 8.3145 J/mol·K × 293 K / ( $2.4 \times 10^3$ L) ≈ 101,325 Pa

Conclusion

In this article, we have explored the relationship between temperature and pressure in a submarine cabin using the ideal gas law. We have provided a numerical setup for calculating the pressure at a given temperature and have demonstrated how to use the ideal gas law to solve for pressure. By understanding the relationship between temperature and pressure, we can ensure the comfort and safety of submarine occupants.

References

Atkins, P. W., & de Paula, J. (2010). Physical chemistry (9th ed.). Oxford University Press.
Chang, R. (2010). Physical chemistry for the life sciences (2nd ed.). W.H. Freeman and Company.

Q: What is the ideal gas law, and how does it relate to submarine cabin pressure?

A: The ideal gas law is a fundamental principle in chemistry that describes the behavior of ideal gases. It states that the product of the pressure (P) and volume (V) of a gas is equal to the product of the number of moles (n) of the gas and the gas constant (R) multiplied by the temperature (T) in Kelvin:

P × V = n × R × T

The ideal gas law is relevant to submarine cabin pressure because it describes how the pressure inside the cabin changes with temperature.

Q: How does the pressure inside a submarine cabin change with temperature?

A: According to the ideal gas law, the pressure inside a submarine cabin increases with temperature. This means that as the temperature inside the cabin increases, the pressure also increases.

Q: What is the relationship between the number of moles of gas and the pressure inside a submarine cabin?

A: The number of moles of gas is directly proportional to the pressure inside a submarine cabin. This means that as the number of moles of gas increases, the pressure inside the cabin also increases.

Q: How can I calculate the pressure inside a submarine cabin at a given temperature?

A: To calculate the pressure inside a submarine cabin at a given temperature, you can use the ideal gas law:

P = n × R × T / V

You will need to know the number of moles of gas (n), the gas constant (R), the temperature (T), and the volume (V) of the cabin.

Q: What are some common applications of the ideal gas law in submarine design?

A: The ideal gas law is used in submarine design to ensure that the pressure inside the cabin is safe and comfortable for occupants. It is also used to calculate the pressure at different depths and temperatures.

Q: Can the ideal gas law be used to calculate the pressure inside a submarine cabin at extreme temperatures?

A: Yes, the ideal gas law can be used to calculate the pressure inside a submarine cabin at extreme temperatures. However, it is essential to ensure that the assumptions of the ideal gas law are valid, such as the gas being ideal and the temperature being in Kelvin.

Q: What are some limitations of the ideal gas law in submarine design?

A: The ideal gas law assumes that the gas is ideal, which is not always the case in real-world applications. Additionally, the ideal gas law does not account for the effects of pressure on the gas, such as compression and expansion.

Q: How can I apply the ideal gas law to real-world submarine design problems?

A: To apply the ideal gas law to real-world submarine design problems, you will need to consider the specific requirements of the submarine, such as the pressure and temperature ranges, and the properties of the gas used. You will also need to ensure that the assumptions of the ideal gas law are valid.

Conclusion

In this article, we have answered some frequently asked questions about submarine cabin pressure and the ideal gas law. We have discussed the relationship between temperature and pressure, the number of moles of gas, and the applications of the ideal gas law in submarine design. By understanding the ideal gas law and its limitations, you can ensure that the pressure inside a submarine cabin is safe and comfortable for occupants.