Gas Pressure And Temperature Problems: Physics Guide

Hey everyone! Let's dive into some interesting problems about gas pressure and temperature. These are classic physics questions that help us understand how gases behave under different conditions. We'll break down each problem step-by-step, so you can grasp the concepts easily. So, let's get started, guys!

1. Calculating Gas Pressure at Different Temperatures

Understanding the Problem: At what temperature will the gas pressure in a cylinder reach $2 \cdot 10^5$ Pa, given that it is initially $1.45 \cdot 10^5$ Pa at 17 °C? This problem involves the relationship between pressure and temperature of a gas when the volume and the amount of gas are kept constant. This is described by Gay-Lussac's Law, which states that the pressure of a gas is directly proportional to its absolute temperature (in Kelvin) when the volume and amount of gas are constant. To solve this, we'll need to convert the given Celsius temperature to Kelvin, set up a proportion using the initial and final conditions, and then solve for the unknown temperature.

Solving the Problem: First, let's convert the initial temperature from Celsius to Kelvin. To do this, we add 273.15 to the Celsius temperature:

$T_1 = 17 °C + 273.15 = 290.15 K$

Now, we can use Gay-Lussac's Law, which can be written as:

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Where:

• $P_1$ is the initial pressure ($1.45 \cdot 10^5$ Pa)
• $T_1$ is the initial temperature (290.15 K)
• $P_2$ is the final pressure ($2 \cdot 10^5$ Pa)
• $T_2$ is the final temperature (unknown)

Plugging in the values, we get:

$\frac{1.45 \cdot 10^5 Pa}{290.15 K} = \frac{2 \cdot 10^5 Pa}{T_2}$

Now, we solve for $T_2$:

$T_2 = \frac{2 \cdot 10^5 Pa \cdot 290.15 K}{1.45 \cdot 10^5 Pa}$

$T_2 = \frac{2 \cdot 290.15}{1.45} K$

$T_2 ≈ 400.21 K$

Finally, let's convert the temperature back to Celsius:

$T_2 = 400.21 K - 273.15 = 127.06 °C$

So, the gas pressure in the cylinder will be equal to $2 \cdot 10^5$ Pa at approximately 127.06 °C.

Key Takeaway: This problem highlights how crucial it is to understand gas laws, especially Gay-Lussac's Law, when dealing with pressure and temperature changes in a closed system. Remember, temperature must always be in Kelvin for these calculations to work correctly. By applying these principles, we can accurately predict how gases will behave under different thermal conditions.

2. Calculating Pressure Increase with Temperature Change

Understanding the Problem: The second question asks, how many times will the gas pressure increase inside a lamp bulb if the temperature rises from 17 °C to 360 °C? This is another application of Gay-Lussac's Law. The key here is to determine the ratio of the final pressure to the initial pressure. We'll follow a similar approach as before: convert Celsius to Kelvin, apply the gas law, and calculate the pressure ratio. Let's break it down.

Solving the Problem: First, we need to convert both temperatures from Celsius to Kelvin:

Initial temperature:

$T_1 = 17 °C + 273.15 = 290.15 K$

Final temperature:

$T_2 = 360 °C + 273.15 = 633.15 K$

Using Gay-Lussac's Law:

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$

We want to find the factor by which the pressure increases, which is $\frac{P_2}{P_1}$. Rearranging the equation, we get:

$\frac{P_2}{P_1} = \frac{T_2}{T_1}$

Now, plug in the Kelvin temperatures:

$\frac{P_2}{P_1} = \frac{633.15 K}{290.15 K}$

$\frac{P_2}{P_1} ≈ 2.18$

Therefore, the gas pressure in the lamp bulb will increase approximately 2.18 times when the temperature rises from 17 °C to 360 °C.

Key Takeaway: This problem reinforces the direct relationship between pressure and temperature as described by Gay-Lussac's Law. It's super important to convert temperatures to Kelvin for accurate calculations. Understanding these relationships helps in practical applications, such as predicting how pressure changes in sealed containers when temperature fluctuates. For example, consider the tire pressure in your car; as the tire heats up during driving, the pressure inside increases, which is a direct application of this principle.

Visualizing the Relationship: Pressure and Temperature Graph

To truly grasp Gay-Lussac's Law, visualizing the relationship between pressure and temperature can be immensely helpful. Imagine plotting pressure (P) on the y-axis and temperature (T) on the x-axis. If we keep the volume and number of moles of the gas constant, the graph will show a straight line passing through the origin. This straight line illustrates the direct proportionality: as temperature increases, pressure increases linearly, and vice versa. Mathematically, this relationship is represented as P = kT, where k is a constant of proportionality. This constant is determined by the volume of the container and the amount of gas present. Understanding this graphical representation not only reinforces the concept but also aids in predicting pressure changes for any given temperature change, and vice versa. It’s a powerful tool in both academic problem-solving and real-world applications, such as designing pressure vessels or understanding meteorological phenomena.

Real-world Applications of Gay-Lussac's Law

Gay-Lussac's Law isn't just a theoretical concept confined to textbooks; it has several practical applications in everyday life and various industries. One prominent example is in the operation and maintenance of vehicle tires. As tires roll on the road, friction causes them to heat up. According to Gay-Lussac's Law, this increase in temperature leads to an increase in tire pressure. Tire manufacturers and car experts advise checking tire pressure when the tires are cold because the pressure readings will be more accurate. Understanding this principle helps drivers maintain correct tire pressure, which is crucial for safety, fuel efficiency, and tire longevity. Overinflated tires can lead to a harsh ride and increased wear in the center of the tire, while underinflated tires can overheat and potentially cause a blowout. Proper tire inflation, guided by Gay-Lussac's Law, ensures optimal vehicle performance and safety.

Another application can be seen in the design of pressure cookers. Pressure cookers work by trapping steam inside a sealed pot. As the water boils and turns into steam, the temperature inside the cooker rises, leading to an increase in pressure. This higher pressure raises the boiling point of water, allowing food to cook faster. The safety valves in pressure cookers are designed to release excess pressure, preventing explosions. The principles of Gay-Lussac's Law are critical in ensuring these cookers operate safely and efficiently. Engineers must consider the pressure-temperature relationship to design cookers that can withstand high pressures and to implement safety mechanisms that prevent hazardous situations. This showcases how fundamental gas laws like Gay-Lussac's Law play a pivotal role in designing practical, everyday devices.

These examples illustrate how a solid grasp of physics, particularly the behavior of gases, is indispensable in numerous practical applications. From ensuring the safety and efficiency of everyday appliances to optimizing vehicle performance, Gay-Lussac's Law helps engineers and consumers alike make informed decisions and design safe systems.

3. (Incomplete Question)

Since the third question is incomplete, we can't provide a solution. However, if you have the full question, feel free to share it, and we'll gladly help you solve it!

Final Thoughts: Understanding the relationship between gas pressure and temperature is fundamental in physics. By applying gas laws like Gay-Lussac's Law, we can solve a variety of problems and understand real-world phenomena. Keep practicing, and you'll master these concepts in no time! Keep those questions coming, guys! We're here to help you learn and grow in your understanding of physics. Happy problem-solving!