Helium gas with a volume of 3.20 3.20 L, under a pressure of 0.180 0.180 atm and at 41.0 41.0 °C, is warmed until both pressure and volume are doubled. What is the final temperature?

everyone in this problem. We have a blue seven times 10 to the negative three m cubed of argon gas. We're told the initial pressure is one atmosphere and the initial temperature is 290 Calvin. The balloon is cooled until both its volume and pressure are halved. And we're asked to find the final temperature. Okay. Alright. So we have some relationship between pressure, temperature and volume. So when we think about that, let's recall the ideal gas equation or the ideal gas law. Okay, that tells us P V is equal to N times R. T. All right. So, we have to kind of situations here, we have the initial scenario and then we have the final scenario. So, let's write out our ideal gas equation for each case. Initially we have P I K. Initial pressure V I. Initial volume and we're gonna divide by the temperature. Okay, initial temperature T I is equal to N times R. Okay, we can do the same thing for the final case. We have P F VF over T F is equal to N times R. All right. Now, when we're thinking about a balloon that has argon gas in it, Okay, this end times our quantity that's going to be constant. Okay, we have the same gas with the same amount of gas. So this is going to be constant. So N R is constant. And what that means is that this P I V I over T I equals N R P F B F over T F equals N R. Okay, so those values must be equal because N times R is equal in both of them. Okay, so this means that P I V I over T I must equal P F VF over TF. Okay, because of the value of N R is equal. Okay, so we can start filling in values here if we want. But let's think about the information we were given first. Okay, we're told that the final volume and the final pressure are half of what the initial pressure and volume are. Okay, so we have P I V I over T I is equal to the final pressure which is one half times the initial pressure. Okay, the final volume which is one half of the initial volume divided by the initial or sorry, the final temperature and again the final temperature T F. That is what we are looking for. Alright, so now we just have an equation with P I V I and T I. Those are all quantities we know okay, we can go ahead and substitute those values and if we want it here. But let's simplify first. Okay, we simplify first. On the left hand side we have P I V. I over T I. And on the right hand side well we just have P I V.I. Okay, over four TF. Let me just get this factor of a quarter. Alright, well now we see that this P I V I if we multiply these will cancel, they were just left with T I. The initial temperature is equal to four times the final temperature. Okay, well this tells us that the final temperature T. F. That we're looking for is one quarter of the initial temperature. And the initial temperature we're given is 200 In 90 Calvin. Which gives us a final temperature of 72.5 Calvin. And so that's that final temperature we were looking for in this case we didn't even need to substitute the value of the pressure and the volume. Okay. We just simplified it first and then we only had a little bit of arithmetic to do there. Okay. Alright, so going back up to the top here, The final temperature that we were looking for is going to be be 72.5 Calvin. That's it for this one. Thanks everyone for watching. See you in the next video.

21. Kinetic Theory of Ideal Gases

The Ideal Gas Law

Physics

21. Kinetic Theory of Ideal Gases

The Ideal Gas Law

4:41 minutes

Problem 2a

Textbook Question

Helium gas with a volume of $3.20$ L, under a pressure of $0.180$ atm and at $41.0$ °C, is warmed until both pressure and volume are doubled. What is the final temperature?

Verified step by step guidance

Start by converting the initial temperature from Celsius to Kelvin. Use the formula: $ T(K) = T(°C) + 273.15 $. For 41.0°C, calculate the temperature in Kelvin.

Apply the ideal gas law in the form $ PV = nRT $ to relate the initial and final states of the gas. Since the number of moles $ n $ and the gas constant $ R $ remain constant, you can use the combined gas law: $ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} $.

Substitute the known values into the combined gas law. The initial pressure $ P_1 = 0.180 $ atm, initial volume $ V_1 = 3.20 $ L, and initial temperature $ T_1 $ in Kelvin. The final pressure $ P_2 = 2 \times 0.180 $ atm and final volume $ V_2 = 2 \times 3.20 $ L.

Rearrange the combined gas law to solve for the final temperature $ T_2 $: $ T_2 = \frac{P_2V_2T_1}{P_1V_1} $.

Substitute the values into the equation and solve for $ T_2 $ to find the final temperature in Kelvin. Remember to check your units to ensure they are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law, PV=nRT, relates the pressure (P), volume (V), and temperature (T) of a gas with its amount in moles (n) and the gas constant (R). It is crucial for understanding how changes in pressure, volume, and temperature affect a gas. In this problem, it helps determine the final temperature when both pressure and volume are doubled.

Temperature Conversion

Temperature conversion is essential when dealing with gas laws, as calculations require temperatures in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. This conversion is necessary to apply the Ideal Gas Law correctly, ensuring accurate calculations of the final temperature.

Direct and Inverse Proportionality

Understanding direct and inverse proportionality helps analyze how variables in the Ideal Gas Law interact. When pressure and volume are doubled, the relationship between these variables and temperature must be considered. Direct proportionality implies that if pressure and volume increase, temperature must also increase to maintain the equation's balance.

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