High Pressures and Temperatures

Oct 13, 2025 by wbrum.org 69 views

Understanding the Problem

When dealing with gases, it's essential to understand the relationship between pressure, volume, and temperature. The ideal gas law, which is a fundamental concept in chemistry, states that the product of the pressure and volume of a gas is directly proportional to the product of the temperature and the number of moles of the gas. Mathematically, this can be expressed as:

PV = nRT

where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.

Given Information

In this problem, we are given the following information:

The gas is stored at a constant volume (V).
The initial pressure (P1) is 137 kPa at a temperature (T1) of 274 K.
The temperature (T2) rises to 296 K.

Applying the Ideal Gas Law

Since the volume is constant, we can rearrange the ideal gas law equation to solve for the pressure (P2) at the new temperature (T2):

P1 / T1 = P2 / T2

We can plug in the given values and solve for P2:

137 kPa / 274 K = P2 / 296 K

Solving for P2

To solve for P2, we can multiply both sides of the equation by 296 K:

P2 = (137 kPa / 274 K) * 296 K

Calculating P2

Now, we can calculate the value of P2:

P2 = 137 kPa * 296 K / 274 K

P2 = 137 kPa * 1.081

P2 = 147.7 kPa

Conclusion

Therefore, if the temperature of the gas rises from 274 K to 296 K at a constant volume, the pressure will increase from 137 kPa to approximately 147.7 kPa.

Real-World Applications

Understanding the relationship between pressure, volume, and temperature is crucial in various real-world applications, such as:

Chemical Engineering: In chemical plants, gases are often compressed or expanded to achieve specific reactions or separations. Understanding the ideal gas law is essential for designing and operating these processes.
Aerospace Engineering: In aircraft and spacecraft, gases are used to propel vehicles or control their trajectory. The ideal gas law is used to calculate the pressure and volume of gases in these applications.
Medical Applications: In medical devices, such as ventilators and oxygen tanks, the ideal gas law is used to calculate the pressure and volume of gases.

Limitations of the Ideal Gas Law

While the ideal gas law is a powerful tool for understanding the behavior of gases, it has some limitations. For example:

Real Gases: Real gases do not behave ideally, and their behavior deviates from the ideal gas law. This is due to the presence of intermolecular forces and the finite size of gas molecules.
High Pressures and Temperatures: At high pressures and temperatures, the ideal gas law is no longer accurate, and more complex equations of state are required to describe the behavior of gases.

Conclusion

In conclusion, the ideal gas law is a fundamental concept in chemistry that describes the relationship between pressure, volume, and temperature. By applying the ideal gas law, we can calculate the pressure of a gas at a constant volume when the temperature changes. This is a crucial concept in various real-world applications, and understanding its limitations is essential for accurate calculations and predictions.

Understanding the Problem

PV = nRT

where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.

Given Information

In this problem, we are given the following information:

The gas is stored at a constant volume (V).
The initial pressure (P1) is 137 kPa at a temperature (T1) of 274 K.
The temperature (T2) rises to 296 K.

Applying the Ideal Gas Law

Since the volume is constant, we can rearrange the ideal gas law equation to solve for the pressure (P2) at the new temperature (T2):

P1 / T1 = P2 / T2

We can plug in the given values and solve for P2:

137 kPa / 274 K = P2 / 296 K

Solving for P2

To solve for P2, we can multiply both sides of the equation by 296 K:

P2 = (137 kPa / 274 K) * 296 K

Calculating P2

Now, we can calculate the value of P2:

P2 = 137 kPa * 296 K / 274 K

P2 = 137 kPa * 1.081

P2 = 147.7 kPa

Conclusion

Therefore, if the temperature of the gas rises from 274 K to 296 K at a constant volume, the pressure will increase from 137 kPa to approximately 147.7 kPa.

Real-World Applications

Understanding the relationship between pressure, volume, and temperature is crucial in various real-world applications, such as:

Chemical Engineering: In chemical plants, gases are often compressed or expanded to achieve specific reactions or separations. Understanding the ideal gas law is essential for designing and operating these processes.
Aerospace Engineering: In aircraft and spacecraft, gases are used to propel vehicles or control their trajectory. The ideal gas law is used to calculate the pressure and volume of gases in these applications.
Medical Applications: In medical devices, such as ventilators and oxygen tanks, the ideal gas law is used to calculate the pressure and volume of gases.

Limitations of the Ideal Gas Law

While the ideal gas law is a powerful tool for understanding the behavior of gases, it has some limitations. For example:

Real Gases: Real gases do not behave ideally, and their behavior deviates from the ideal gas law. This is due to the presence of intermolecular forces and the finite size of gas molecules.
High Pressures and Temperatures: At high pressures and temperatures, the ideal gas law is no longer accurate, and more complex equations of state are required to describe the behavior of gases.

Q&A

Q: What is the ideal gas law?

A: The ideal gas law is a fundamental concept in chemistry that describes the relationship between pressure, volume, and temperature of a gas. It is expressed mathematically as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.

Q: What is the relationship between pressure and temperature in the ideal gas law?

A: According to the ideal gas law, the pressure of a gas is directly proportional to the temperature, assuming a constant volume.

Q: What is the effect of increasing the temperature of a gas at constant volume on its pressure?

A: Increasing the temperature of a gas at constant volume will increase its pressure.

Q: What is the effect of increasing the volume of a gas at constant temperature on its pressure?

A: Increasing the volume of a gas at constant temperature will decrease its pressure.

Q: What are some real-world applications of the ideal gas law?

A: The ideal gas law is used in various real-world applications, such as chemical engineering, aerospace engineering, and medical applications.

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

A: The ideal gas law has some limitations, such as the behavior of real gases, which do not behave ideally, and high pressures and temperatures, where more complex equations of state are required.

Q: How can the ideal gas law be used to calculate the pressure of a gas at a constant volume when the temperature changes?

A: The ideal gas law can be used to calculate the pressure of a gas at a constant volume when the temperature changes by rearranging the equation to solve for the pressure (P2) at the new temperature (T2): P1 / T1 = P2 / T2.

Q: What is the value of P2 in the given problem?

A: The value of P2 is approximately 147.7 kPa.

Q: What is the effect of increasing the temperature of the gas from 274 K to 296 K at a constant volume on its pressure?

A: The pressure of the gas will increase from 137 kPa to approximately 147.7 kPa.