A water molecule has its three atoms arranged in a 'V' shape, so it has rotational kinetic energy around any of three mutually perpendicular axes. However, like diatomic molecules, its vibrational modes are not active at temperatures below 1000 K. What is the thermal energy of 2.0 mol of steam at a temperature of 160°C?

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Hey, everyone. Let's go through this practice problem. A hypothetical gas with four atoms bonded in A T shape has a high bond energy that requires temperatures greater or equal to 1200 Kelvins to activate vibrational modes. The hypothetical gas has rotational kinetic energy in a 3d coordinate system at all. Temperatures calculate the thermal energy of a sample of the gas with N equals 6.2 moles at T equals 350 °C option A 48.2 kilojoules. Option B 145 kilojoules, option C 96.3 kilojoules, option D 80.2 kilojoules and option E 15.5 kilojoules. Since we're looking for the thermal energy of the sample. Let's recall the equi partition theorem which states that the thermal energy is distributed throughout the sample of gas such that each degree of freedom contributes a quantity of energy of one half multiplied by N RT. The number of moles multiplied by the ideal gas constant multiplied by the temperatures. The the gas is temperature. So we must first establish the number of degrees of freedom that the gas has. As pretty much always, there are going to be the three degrees of freedom for translational motion. Since we're looking at a three dimensional space, but there are also going to be three degrees of freedom from rotational motion. A lot of the times there are only two if we're looking at a diatomic molecule. But since the problem specifically states that we're analyzing a hypothetical gas with four atoms, then it can rotate in three degrees. So there are six degrees of freedom in total. Let's also talk about the temperature real quick since the temperature is given in the problem as 350 °C. But in order for us to use this in the thermal equations, we need to convert this from degrees Celsius into Kelvin's by adding 273. So the temperature of the gas sample is equal to 623 Kelvins. Now let's plug these variables into the equation. So first off, as mentioned earlier, the energy equation that we've written down is only for one degree of freedom. But now that we know that we're looking at six degrees of freedom, we multiply the whole thing by six. So six divided by two, multiplied by the number of moles, 6.2 moles multiplied by the ideal gas constant, 8.314 joules per mole. Kelvins multiplied by the temperature of 623 Kelvins. And if we put that into a calculator, and we find in that amount of energy about 96,341 joules or if we round this down to three significant figures, as all the multiple choice options are in, then we can write this as 96.3 Hela Jules. So 96.3 kilojoules is our answer to the problem. And if you look at our multiple choice options, this agrees with option C. So option C is the correct answer to the problem and that is it for this problem? I hope this video helped you out. If you think 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. I hope you all have a good day.

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Key Concepts

Thermal Energy

Ideal Gas Law

Molecular Degrees of Freedom