The Pressure In A Car Tire Is 2.47 Atm At $27.6^ \circ} C$. What Will The Pressure Be If The Temperature Increases To $35.7^{\circ} C$?Note $T(K) = T\left({ ^{\circ} C \right) + 273.15$ □ \square □ Atm

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Introduction

The pressure in a car tire is a critical factor that affects its performance, fuel efficiency, and overall safety. The pressure is measured in atmospheres (atm) and is influenced by various factors, including temperature. In this article, we will explore how temperature change affects the pressure in a car tire, using a specific example to illustrate the concept.

The Ideal Gas Law

The ideal gas law is a fundamental principle in physics that describes the behavior of gases under different conditions. It states that the pressure (P) of a gas is directly proportional to its temperature (T) and inversely proportional to its volume (V). Mathematically, this can be expressed as:

P = k * T / V

where k is a constant that depends on the gas and its properties.

Converting Temperature from Celsius to Kelvin

To apply the ideal gas law, we need to convert the temperature from Celsius to Kelvin. The formula for this conversion is:

T(K) = T($\left({ }^{\circ} C \right) + 273.15$

Using this formula, we can convert the initial temperature of 27.6°C to Kelvin:

T(K) = 27.6°C + 273.15 = 300.75 K

Calculating the Initial Pressure

The initial pressure in the car tire is given as 2.47 atm at a temperature of 27.6°C. We can use the ideal gas law to calculate the pressure at this temperature, assuming a constant volume.

P = k * T / V

Since we are not given the value of k, we can assume it to be a constant for this example. Therefore, we can write:

P1 = k * T1 / V

where P1 is the initial pressure, T1 is the initial temperature, and V is the volume.

Calculating the Final Pressure

Now, let's calculate the final pressure at a temperature of 35.7°C. First, we need to convert this temperature to Kelvin:

T(K) = 35.7°C + 273.15 = 308.85 K

Using the ideal gas law, we can write:

P2 = k * T2 / V

where P2 is the final pressure, T2 is the final temperature, and V is the volume.

Applying the Gas Law

Since the volume is constant, we can cancel it out from both equations:

P1 / T1 = P2 / T2

Rearranging this equation, we get:

P2 = P1 * T2 / T1

Substituting the values, we get:

P2 = 2.47 atm * 308.85 K / 300.75 K

Calculating the Final Pressure

Now, let's calculate the final pressure:

P2 = 2.47 atm * 308.85 K / 300.75 K P2 = 2.53 atm

Conclusion

In conclusion, we have used the ideal gas law to calculate the pressure in a car tire at a temperature of 35.7°C, given the initial pressure and temperature. The final pressure is 2.53 atm, which is higher than the initial pressure due to the increase in temperature. This example illustrates the importance of considering temperature changes when working with gases and their properties.

References

Discussion

Q: What is the ideal gas law, and how does it relate to the pressure in a car tire?

A: The ideal gas law is a fundamental principle in physics that describes the behavior of gases under different conditions. It states that the pressure (P) of a gas is directly proportional to its temperature (T) and inversely proportional to its volume (V). Mathematically, this can be expressed as:

P = k * T / V

where k is a constant that depends on the gas and its properties. In the context of a car tire, the ideal gas law helps us understand how temperature changes affect the pressure inside the tire.

Q: Why is it important to consider temperature changes when working with gases and their properties?

A: Temperature changes can significantly affect the pressure of a gas. In the case of a car tire, an increase in temperature can cause the air molecules to expand and increase the pressure inside the tire. Conversely, a decrease in temperature can cause the air molecules to contract and decrease the pressure inside the tire. This is why it's essential to consider temperature changes when working with gases and their properties.

Q: How does the ideal gas law apply to a car tire in different scenarios?

A: The ideal gas law applies to a car tire in various scenarios, including:

  • Temperature changes: As we discussed earlier, temperature changes can affect the pressure inside a car tire. The ideal gas law helps us understand how temperature changes impact the pressure.
  • Altitude changes: As altitude increases, the atmospheric pressure decreases. This can affect the pressure inside a car tire, especially if the tire is not designed to handle the lower pressure.
  • Volume changes: If the volume of a car tire changes, the pressure inside the tire will also change. This can occur if the tire is inflated or deflated.

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

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

  • Aerospace engineering: The ideal gas law is used to calculate the pressure and temperature of gases in spacecraft and aircraft.
  • Chemical engineering: The ideal gas law is used to design and optimize chemical processes, such as those involved in the production of fuels and chemicals.
  • Medical devices: The ideal gas law is used to design and optimize medical devices, such as ventilators and oxygen tanks.

Q: How can I calculate the pressure in a car tire using the ideal gas law?

A: To calculate the pressure in a car tire using the ideal gas law, you will need to know the following:

  • Initial pressure: The initial pressure of the tire in atm.
  • Initial temperature: The initial temperature of the tire in °C.
  • Final temperature: The final temperature of the tire in °C.
  • Constant k: The constant k depends on the gas and its properties. For air, k is approximately 0.0821 L atm/mol K.

Using the ideal gas law, you can calculate the final pressure of the tire as follows:

P2 = P1 * T2 / T1

where P2 is the final pressure, P1 is the initial pressure, T2 is the final temperature, and T1 is the initial temperature.

Q: What are some common mistakes to avoid when working with the ideal gas law?

A: Some common mistakes to avoid when working with the ideal gas law include:

  • Not considering temperature changes: Temperature changes can significantly affect the pressure of a gas. Make sure to consider temperature changes when working with the ideal gas law.
  • Not using the correct units: Make sure to use the correct units when working with the ideal gas law. For example, use atm for pressure and °C for temperature.
  • Not accounting for volume changes: If the volume of a gas changes, the pressure will also change. Make sure to account for volume changes when working with the ideal gas law.

Conclusion

In conclusion, the ideal gas law is a fundamental principle in physics that describes the behavior of gases under different conditions. It has numerous real-world applications, including aerospace engineering, chemical engineering, and medical devices. By understanding the ideal gas law and its applications, you can better design and optimize systems that involve gases and their properties.