A $475 \, \text{cm}^3$ Balloon Of Gas At Standard Temperature And Pressure Warms In The Sun And Expands Until It Occupies A Volume Of $600 \, \text{cm}^3$. What Temperature Is The Gas At Now? (Assume $\frac{PV}{T}$ Is

Understanding the Problem

To solve this problem, we need to apply the ideal gas law, which 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.

However, we are given the equation $\frac{PV}{T}$ is constant, which is a more general form of the ideal gas law. This equation can be rearranged to get $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$, where the subscripts 1 and 2 refer to the initial and final states of the gas, respectively.

Given Information

We are given the following information:

• The initial volume of the gas is $475 \, \text{cm}^3$.
• The final volume of the gas is $600 \, \text{cm}^3$.
• The gas is at standard temperature and pressure (STP) initially, which means that the initial temperature is $273.15 \, \text{K}$ and the initial pressure is $101.325 \, \text{kPa}$.
• We assume that the pressure remains constant during the expansion.

Applying the Ideal Gas Law

We can now apply the ideal gas law to solve for the final temperature of the gas. We can rearrange the equation $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ to get $\frac{T_2}{T_1} = \frac{P_2V_2}{P_1V_1}$.

Since the pressure remains constant, we can simplify the equation to get $\frac{T_2}{T_1} = \frac{V_2}{V_1}$.

Calculating the Final Temperature

We can now plug in the values to calculate the final temperature of the gas.

• The initial volume is $475 \, \text{cm}^3$.
• The final volume is $600 \, \text{cm}^3$.
• The initial temperature is $273.15 \, \text{K}$.

We can now calculate the final temperature using the equation $\frac{T_2}{T_1} = \frac{V_2}{V_1}$.

Solution

We can now solve for the final temperature of the gas.

$\frac{T_2}{273.15} = \frac{600}{475}$

$T_2 = \frac{600}{475} \times 273.15$

$T_2 = 303.15 \, \text{K}$

Conclusion

The final temperature of the gas is $303.15 \, \text{K}$.

Standard Temperature and Pressure (STP)

Standard temperature and pressure (STP) is a set of conditions that are used as a reference point for the measurement of physical properties of gases. The conditions are:

• Temperature: $273.15 \, \text{K}$ (or $0 \, \text{°C}$ or $32 \, \text{°F}$)
• Pressure: $101.325 \, \text{kPa}$ (or $1 \, \text{atm}$ or $760 \, \text{mmHg}$)

Ideal Gas Law

The ideal gas law is a mathematical equation that describes the behavior of ideal gases. The equation is:

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

Gas Constant

The gas constant is a physical constant that is used in the ideal gas law. The value of the gas constant is:

R = 8.3145 , \text{J/mol·K}$

Conclusion

In conclusion, the final temperature of the gas is $303.15 \, \text{K}$. This is a result of the gas expanding from an initial volume of $475 \, \text{cm}^3$ to a final volume of $600 \, \text{cm}^3$ at constant pressure.

Understanding the Problem

To solve this problem, we need to apply the ideal gas law, which 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.

However, we are given the equation $\frac{PV}{T}$ is constant, which is a more general form of the ideal gas law. This equation can be rearranged to get $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$, where the subscripts 1 and 2 refer to the initial and final states of the gas, respectively.

Given Information

We are given the following information:

• The initial volume of the gas is $475 \, \text{cm}^3$.
• The final volume of the gas is $600 \, \text{cm}^3$.
• The gas is at standard temperature and pressure (STP) initially, which means that the initial temperature is $273.15 \, \text{K}$ and the initial pressure is $101.325 \, \text{kPa}$.
• We assume that the pressure remains constant during the expansion.

Applying the Ideal Gas Law

We can now apply the ideal gas law to solve for the final temperature of the gas. We can rearrange the equation $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ to get $\frac{T_2}{T_1} = \frac{P_2V_2}{P_1V_1}$.

Since the pressure remains constant, we can simplify the equation to get $\frac{T_2}{T_1} = \frac{V_2}{V_1}$.

Calculating the Final Temperature

We can now plug in the values to calculate the final temperature of the gas.

• The initial volume is $475 \, \text{cm}^3$.
• The final volume is $600 \, \text{cm}^3$.
• The initial temperature is $273.15 \, \text{K}$.

We can now calculate the final temperature using the equation $\frac{T_2}{T_1} = \frac{V_2}{V_1}$.

Solution

We can now solve for the final temperature of the gas.

$\frac{T_2}{273.15} = \frac{600}{475}$

$T_2 = \frac{600}{475} \times 273.15$

$T_2 = 303.15 \, \text{K}$

Conclusion

The final temperature of the gas is $303.15 \, \text{K}$.

Q&A

Q: What is the ideal gas law?

A: The ideal gas law is a mathematical equation that describes the behavior of ideal gases. The equation is 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 gas constant?

A: The gas constant is a physical constant that is used in the ideal gas law. The value of the gas constant is R = 8.3145 , \text{J/mol·K}$.

Q: What is standard temperature and pressure (STP)?

A: Standard temperature and pressure (STP) is a set of conditions that are used as a reference point for the measurement of physical properties of gases. The conditions are:

• Temperature: $273.15 \, \text{K}$ (or $0 \, \text{°C}$ or $32 \, \text{°F}$)
• Pressure: $101.325 \, \text{kPa}$ (or $1 \, \text{atm}$ or $760 \, \text{mmHg}$)

Q: How do I calculate the final temperature of a gas that expands from an initial volume to a final volume at constant pressure?

A: To calculate the final temperature of a gas that expands from an initial volume to a final volume at constant pressure, you can use the equation $\frac{T_2}{T_1} = \frac{V_2}{V_1}$, where T1 and T2 are the initial and final temperatures, and V1 and V2 are the initial and final volumes.

Q: What is the final temperature of a gas that expands from an initial volume of $475 \, \text{cm}^3$ to a final volume of $600 \, \text{cm}^3$ at constant pressure?

A: The final temperature of the gas is $303.15 \, \text{K}$.

Q: What is the relationship between the pressure and volume of a gas?

A: The pressure and volume of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the pressure of a gas is directly proportional to the volume of the gas, provided that the temperature and the number of moles of the gas remain constant.

Q: What is the relationship between the temperature and volume of a gas?

A: The temperature and volume of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the temperature of a gas is directly proportional to the volume of the gas, provided that the pressure and the number of moles of the gas remain constant.

Q: What is the relationship between the number of moles and the volume of a gas?

A: The number of moles and the volume of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the number of moles of a gas is directly proportional to the volume of the gas, provided that the pressure and the temperature of the gas remain constant.

Q: What is the relationship between the pressure and temperature of a gas?

A: The pressure and temperature of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the pressure of a gas is directly proportional to the temperature of the gas, provided that the volume and the number of moles of the gas remain constant.

Q: What is the relationship between the volume and temperature of a gas?

A: The volume and temperature of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the volume of a gas is directly proportional to the temperature of the gas, provided that the pressure and the number of moles of the gas remain constant.

Q: What is the relationship between the number of moles and the temperature of a gas?

A: The number of moles and the temperature of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the number of moles of a gas is directly proportional to the temperature of the gas, provided that the pressure and the volume of the gas remain constant.

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

A: The pressure and the number of moles of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the pressure of a gas is directly proportional to the number of moles of the gas, provided that the volume and the temperature of the gas remain constant.

Q: What is the relationship between the volume and the number of moles of a gas?

A: The volume and the number of moles of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the volume of a gas is directly proportional to the number of moles of the gas, provided that the pressure and the temperature of the gas remain constant.

Q: What is the relationship between the temperature and the number of moles of a gas?

A: The temperature and the number of moles of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the temperature of a gas is directly proportional to the number of moles of the gas, provided that the pressure and the volume of the gas remain constant.

Q: What is the relationship between the pressure, volume, and temperature of a gas?

A: The pressure, volume, and temperature of a gas are related by the ideal gas law, which is PV = nRT. This equation shows that the pressure of a gas is directly proportional to the product of the volume and the temperature of the gas, provided that the number of moles of the gas remains constant.

Q: What is the relationship between the volume, temperature, and number of moles of a gas?

A: The volume, temperature, and number of moles of a gas are related by the ideal gas law, which is **PV