Calculating the Initial Temperature

Oct 13, 2025 by thesweetdesigns.com 73 views

Introduction

In this article, we will explore the concept of gas mixtures and how to calculate the change in temperature of a gas mixture when a new gas is added to a rigid container. We will use the ideal gas law to solve this problem and understand the relationship between the number of moles, pressure, and temperature of a gas mixture.

The Ideal Gas Law

The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases. It is given by the equation:

PV = nRT

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

Initial Conditions

We are given a rigid 2.50 L bottle containing 0.458 mol of helium (He) gas at a pressure of 1.83 atm. We can use the ideal gas law to calculate the initial temperature of the gas mixture.

Calculating the Initial Temperature

We can rearrange the ideal gas law to solve for temperature:

T = PV / nR

We know the initial pressure (P1 = 1.83 atm), the initial number of moles (n1 = 0.458 mol), and the volume (V = 2.50 L). We can plug in these values and solve for the initial temperature (T1).

import math

# Given values
P1 = 1.83  # atm
n1 = 0.458  # mol
V = 2.50  # L
R = 0.08206  # L atm/mol K

# Calculate the initial temperature
T1 = (P1 * V) / (n1 * R)
print(f"The initial temperature is {T1:.2f} K")

Adding Argon to the Bottle

We are given that 0.713 mol of argon (Ar) is added to the bottle, and the pressure increases to 2.05 atm. We can use the ideal gas law to calculate the final temperature of the gas mixture.

Calculating the Final Temperature

We know the final pressure (P2 = 2.05 atm), the final number of moles (n2 = n1 + 0.713 mol = 1.171 mol), and the volume (V = 2.50 L). We can plug in these values and solve for the final temperature (T2).

# Given values
P2 = 2.05  # atm
n2 = n1 + 0.713  # mol

# Calculate the final temperature
T2 = (P2 * V) / (n2 * R)
print(f"The final temperature is {T2:.2f} K")

Calculating the Change in Temperature

We can calculate the change in temperature by subtracting the initial temperature from the final temperature:

ΔT = T2 - T1

# Calculate the change in temperature
dT = T2 - T1
print(f"The change in temperature is {dT:.2f} K")

Conclusion

In this article, we used the ideal gas law to calculate the change in temperature of a gas mixture when a new gas is added to a rigid container. We first calculated the initial temperature of the gas mixture using the ideal gas law, and then calculated the final temperature after adding argon to the bottle. Finally, we calculated the change in temperature by subtracting the initial temperature from the final temperature.

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.

Note

Introduction

In our previous article, we explored the concept of gas mixtures and how to calculate the change in temperature of a gas mixture when a new gas is added to a rigid container. We used the ideal gas law to solve this problem and understand the relationship between the number of moles, pressure, and temperature of a gas mixture. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the ideal gas law?

A: The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases. It is given by the equation:

PV = nRT

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

Q: What is the significance of the gas constant (R)?

A: The gas constant (R) is a fundamental constant in chemistry that relates the pressure, volume, and temperature of a gas. It is approximately equal to 0.08206 L atm/mol K.

Q: How do I calculate the initial temperature of a gas mixture?

A: To calculate the initial temperature of a gas mixture, you can use the ideal gas law and rearrange it to solve for temperature:

T = PV / nR

You can plug in the values of pressure, volume, and number of moles to solve for the initial temperature.

Q: What happens when a new gas is added to a rigid container?

A: When a new gas is added to a rigid container, the pressure of the gas mixture increases. The ideal gas law can be used to calculate the final temperature of the gas mixture.

Q: How do I calculate the change in temperature of a gas mixture?

A: To calculate the change in temperature of a gas mixture, you can subtract the initial temperature from the final temperature:

ΔT = T2 - T1

Q: What are some common mistakes to avoid when calculating the change in temperature of a gas mixture?

A: Some common mistakes to avoid when calculating the change in temperature of a gas mixture include:

Not using the correct values for pressure, volume, and number of moles
Not using the correct value for the gas constant (R)
Not accounting for the change in pressure when a new gas is added to the container
Not using the ideal gas law to calculate the initial and final temperatures

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

A: The ideal gas law has many real-world applications, including:

Calculating the pressure and temperature of gases in industrial processes
Designing and optimizing gas storage tanks and containers
Understanding the behavior of gases in biological systems
Calculating the change in temperature of gases in various engineering applications

Conclusion

In this article, we answered some frequently asked questions related to calculating the change in temperature of a gas mixture when a new gas is added to a rigid container. We hope that this article has provided you with a better understanding of the ideal gas law and its applications.

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.

Note

The ideal gas law is a simplified model that assumes ideal behavior of gases. In reality, gases do not behave ideally, and the ideal gas law is only an approximation. However, for the purposes of this problem, we will assume ideal behavior and use the ideal gas law to calculate the change in temperature of the gas mixture.