A gas of 1.0 x 10²⁰ atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?

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Hello everyone. Let's go through this problem. A container is filled with 2.4, multiplied by 10 to the power of 19 molecules of a gaseous substance. The heat energy of the gas in the container is 0.80 Jews. If C sub P, the specific heat at constant pressure of the gas is 29.1. Joules per mole. Kelvins determine the temperature of the gas option. A 690 Kelvins option B 69 Kelvins option C 96.6 calvins and option D 966 calvins. When we're dealing with a relationship between temperature, a thermal energy and specific heats, there is a very useful equation to know and it states that the thermal energy or a gas is equal to the number of moles of that gas multiplied by the specific heat at constant volume multiplied by the temperature of the gas. The temperature is what we want to find. So let's algebraically rewrite this equation to sulfur temperature by dividing both sides of the equation by N C sub V. And we find that the temperature is equal to the total thermal energy divided by the number of moles multiplied by the specific heat at constant volume. There are a couple of things we need to establish. First. However, first of all, we don't know the specific heat at constant volume because the problem only gives us the specific heat at constant pressure. C sub P. Fortunately, though there's a pretty simple relationship that we have between the two different specific heats and the ideal gas constant. So the specific heat, a constant pressure is equal to the specific heat at constant volume plus the ideal gas constant. So if we want to solve for the specific heat at constant volume, we simply subtract from both sides of the equation, the ideal gas constant. And we find that the specific heat at constant volume is equal to the specific heat at constant pressure minus the ideal gas constant. Or for us, that's 29.1 jewels per mole. Kelvins minus 8.314 jewels per mole calvins. And if we put this into a calculator, then we find a specific heat at constant volume of about 20.786 jewels per mole calvins. The next thing we need to establish is the number of moles of the gas. We're given the actual number of molecules in the gas. But in order to use the equation, we mentioned earlier, we need to have the number of mold. Fortunately, there's a pretty simple formula for that as well. Recall that the number of moles of a substance is equal to the number of particles of that thing divided by the aga constant because the avogadro constant gives the number of objects per mole, the number of molecules per mole for a gas. So for us, the number of molecules has been given to us as 2.4 multiplied by 10 to the power of 19 molecules. And we're dividing that by the avogadro constant. And recall that the avogadro number is 6. multiplied by 10 to the power of 23. And the units are molecules per mole. If you put this into a calculator, we find a number of moles of about 3.985 multiplied by 10 to the power of negative five moles. All that's up for us to do now is plug the numbers we found into the original equation and put it into a calculator to solve for the temperature. So the temperature is equal to the thermal energy, which is given to us in the problem as 0.8 Jews. And this is being divided by the number of moles 3.985, multiplied by 10 of the power of negative five moles and is being multiplied by the specific heat at constant volume or 20.786 jewels per mole calvins. If we put this into a calculator, then we find a temperature of approximately 966 Kelvins. So that is our answer to this problem. If you look at our multiple choice options, this agrees with option D 966 Kelvins. So option D is the correct answer to this problem and that's it for this video. I hope this video helped you out. If you have. If you need more practice, please check out some of our other tutoring videos which will give you more experience with these types of problems, but that's all for now. and I hope you all have a lovely day. Bye bye.

A gas of 1.0 x 10²⁰ atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?

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Key Concepts

Thermal Energy

Molar Specific Heat

Ideal Gas Law