A Gas In A Rigid Container At $25^{\circ} C$ Has A Pressure Of 0.96 Atm. A Change In Temperature Causes The Pressure To Increase To 1.25 Atm. What Is The New Temperature Of The Gas?Use $\frac{P_1}{T_1}=\frac{P_2}{T_2}$.A.
The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases under various conditions. It is a crucial tool for understanding the properties and behavior of gases, and it has numerous applications in various fields, including chemistry, physics, and engineering. In this article, we will explore the ideal gas law and its applications, with a focus on a specific problem involving a gas in a rigid container.
The Ideal Gas Law
The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, and temperature of a gas. It is expressed as:
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
- T is the temperature of the gas in Kelvin
The Problem
A gas in a rigid container at $25^{\circ} C$ has a pressure of 0.96 atm. A change in temperature causes the pressure to increase to 1.25 atm. What is the new temperature of the gas?
Using the Ideal Gas Law to Solve the Problem
To solve this problem, we can use the ideal gas law and the given information to find the new temperature of the gas. We are given the initial pressure, temperature, and final pressure, and we need to find the final temperature.
We can start by using the ideal gas law to express the relationship between the pressure and temperature of the gas:
PV = nRT
Since the volume of the gas is constant (it is in a rigid container), we can rearrange the equation to get:
P1 / T1 = P2 / T2
where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature.
Solving for the New Temperature
We are given the following values:
- P1 = 0.96 atm
- T1 = 25°C = 298 K
- P2 = 1.25 atm
- T2 = ? (we need to find this value)
We can plug these values into the equation and solve for T2:
P1 / T1 = P2 / T2
(0.96 atm) / (298 K) = (1.25 atm) / T2
To solve for T2, we can cross-multiply and rearrange the equation:
T2 = (1.25 atm) * (298 K) / (0.96 atm)
T2 = 323.4 K
Converting the Temperature to Celsius
To convert the temperature from Kelvin to Celsius, we can use the following equation:
T (°C) = T (K) - 273.15
T2 (°C) = 323.4 K - 273.15
T2 (°C) = 50.25°C
Conclusion
In this article, we used the ideal gas law to solve a problem involving a gas in a rigid container. We were given the initial pressure, temperature, and final pressure, and we needed to find the final temperature. We used the equation P1 / T1 = P2 / T2 to solve for the new temperature, and we found that the new temperature is 50.25°C.
Applications of the Ideal Gas Law
The ideal gas law has numerous applications in various fields, including chemistry, physics, and engineering. Some of the key applications include:
- Calculating the pressure of a gas: The ideal gas law can be used to calculate the pressure of a gas at a given temperature and volume.
- Calculating the volume of a gas: The ideal gas law can be used to calculate the volume of a gas at a given pressure and temperature.
- Calculating the temperature of a gas: The ideal gas law can be used to calculate the temperature of a gas at a given pressure and volume.
- Understanding the behavior of gases: The ideal gas law can be used to understand the behavior of gases under various conditions, including changes in pressure, temperature, and volume.
Real-World Examples
The ideal gas law has numerous real-world applications, including:
- Airbags: The ideal gas law is used to design airbags that deploy quickly and safely in the event of a crash.
- Scuba diving: The ideal gas law is used to calculate the pressure of the air in a scuba tank and to ensure that the diver is not exposed to too much pressure.
- Industrial processes: The ideal gas law is used to design and optimize industrial processes, such as the production of chemicals and the separation of mixtures.
- Atmospheric science: The ideal gas law is used to understand the behavior of the atmosphere and to predict changes in weather patterns.
Conclusion
The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases under various conditions. In this article, we will answer some of the most frequently asked questions about the ideal gas law.
Q: What is the ideal gas law?
A: The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, and temperature of a gas. It is expressed as:
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
- T is the temperature of the gas in Kelvin
Q: What are the assumptions of the ideal gas law?
A: The ideal gas law assumes that the gas is:
- Ideal: The gas is assumed to be an ideal gas, which means that it has no intermolecular forces and no volume.
- Perfectly elastic: The gas is assumed to be perfectly elastic, which means that it can be compressed and expanded without any loss of energy.
- Non-reactive: The gas is assumed to be non-reactive, which means that it does not react with other substances.
Q: What are the limitations of the ideal gas law?
A: The ideal gas law has several limitations, including:
- Real gases are not ideal: Real gases are not ideal and have intermolecular forces and volume.
- Temperature and pressure limitations: The ideal gas law is only valid at low temperatures and pressures.
- Non-ideal behavior: The ideal gas law does not account for non-ideal behavior, such as the behavior of gases at high pressures and temperatures.
Q: How is the ideal gas law used in real-world applications?
A: The ideal gas law is used in a wide range of real-world applications, including:
- Airbags: The ideal gas law is used to design airbags that deploy quickly and safely in the event of a crash.
- Scuba diving: The ideal gas law is used to calculate the pressure of the air in a scuba tank and to ensure that the diver is not exposed to too much pressure.
- Industrial processes: The ideal gas law is used to design and optimize industrial processes, such as the production of chemicals and the separation of mixtures.
- Atmospheric science: The ideal gas law is used to understand the behavior of the atmosphere and to predict changes in weather patterns.
Q: What are some common mistakes to avoid when using the ideal gas law?
A: Some common mistakes to avoid when using the ideal gas law include:
- Not accounting for non-ideal behavior: The ideal gas law does not account for non-ideal behavior, such as the behavior of gases at high pressures and temperatures.
- Not using the correct units: The ideal gas law requires the use of the correct units, including the gas constant and the temperature in Kelvin.
- Not considering the limitations of the ideal gas law: The ideal gas law has several limitations, including the assumption of ideal behavior and the limitations of temperature and pressure.
Q: How can I apply the ideal gas law to a real-world problem?
A: To apply the ideal gas law to a real-world problem, you can follow these steps:
- Identify the problem: Identify the problem you are trying to solve and the variables involved.
- Choose the correct equation: Choose the correct equation from the ideal gas law, such as PV = nRT.
- Plug in the values: Plug in the values for the variables, including the pressure, volume, number of moles, and temperature.
- Solve for the unknown: Solve for the unknown variable, such as the pressure or temperature.
- Check the units: Check the units to ensure that they are correct.
Q: What are some common applications of the ideal gas law?
A: Some common applications of the ideal gas law include:
- Airbags: The ideal gas law is used to design airbags that deploy quickly and safely in the event of a crash.
- Scuba diving: The ideal gas law is used to calculate the pressure of the air in a scuba tank and to ensure that the diver is not exposed to too much pressure.
- Industrial processes: The ideal gas law is used to design and optimize industrial processes, such as the production of chemicals and the separation of mixtures.
- Atmospheric science: The ideal gas law is used to understand the behavior of the atmosphere and to predict changes in weather patterns.
Conclusion
In conclusion, the ideal gas law is a fundamental concept in chemistry that describes the behavior of gases under various conditions. It has numerous applications in various fields, including chemistry, physics, and engineering. In this article, we answered some of the most frequently asked questions about the ideal gas law and provided examples of its use in real-world applications.