A 6.0 m ✕ 8.0 m ✕ 3.0 m room contains air at 20℃. What is the room's thermal energy?

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Hi, everyone in this practice problem. We're being asked to calculate the total thermal energy of a room. We'll have an art gallery with a cubic room with sites measuring 10 m each maintaining a consistent air temperature of 34 degrees Celsius. The options given for the total thermal energy of the room R A 6.97 times 10 to the power of seven Jules, B 4.12 times 10 to the power of six jules, C 2.53 times 10 to the power of eight Jules. And lastly D 3.31 times 10 to the power of seven jos. So the thermal energy of the room can be calculated by using the equation of E P H equals to N multiplied by Z C P multiplied by T. We call, we will call this equation one where in this case E P H is the thermal energy N is the number of moles C P is the molar specific heat and T is the temperature. So the molar specific heat of a diatomic gas or C P can be calculated by C P equals to five divided by two multiplied by R. So R is just a constant R and a gas constant, which is going to be equals to 8. jules per or jules divided by K Cain mole. So to calculate the thermal energy, we need the CV value. So let's calculate that. So CV will then be equals to five divided by two, multiplied by 8.314 jules divided by Calvin mole. And then calculating this, we will then get our CV value of 20.8 jules divided by Calvin mole. Next, what we wanna uh find after this, after CV is the N or the number of mole value. We can uh assume that the gas inside of the room is going to be ideal. So we can use the ideal gas equation where P multiplied by V will equals to N multiplied by R multiplied by T. And what we're interested to find is the total number of moles, which is N. So you wanna rearrange our ideal guess equation to get an equation for N which will then be equals to PV divided by RT. So let's actually um break each of this variable down. So P is going to be the atmospheric pressure. So the atmospheric pressure P will equals to 1.13 times 10 to the power of five pascal or one ATM. And then V is the volume of the cubic room which in this case, V will then equals to 10 m, multiplied by 10 m and then multiplied by 10 m again, for the last, the last site calculating this, the volume will then be equals to 1000 m cube. The R is still going to equal to our uh gas constant, which is 8.314 joules per Calvin mole. And lastly T is the temperature in Kelvin, where T is given in the problem statement to be 24 degrees Celsius. So converting 24 degrees Celsius into Calvin, we have to multi uh we have to plus that add this with 273 Calvin. So this will give us our temperature value of 297 Calvin. So then we can actually substitute all of these values inside of our equation so that our N or mole will then equals to 1.13 times 10 to the power of five pascals multiply that by the volume, which is 1000 m cube divide that with the gas constant, which is 8.314 jules divided by Calvin mole and then multiplied that lastly with the temperature which is 297. Kelvin calculating all of this, we will get our total number of moles to then be 41, 0 to 4 or 41, 40.5 mole. Awesome. So then we know what the N or the number of mole is and we know what the CV is. And lastly, we also know what our temperature is. So we can utilize our equation here for thermal energy to calculate the thermal energy of the room. So E P H will equals to N multiplied by CV, multiplied by T which will equals to N which is 41,024 0.5 more multiplied that by CV, that we just calculated which is 20.8 joles divided by Cain mole divided by kin mo. And then multiply that with our temperature in Calvin, which is going to be 297 Kelvin. After doing our calculation, we will then get our E P H value to then be 2.53 times to the power of eight JOS. So that will be the final thermal energy of the room which is 2. times 10 to the power of AJ which will correspond to option C in our answer choices. So option C will be the answer to this particular practice problem and that will be it for this video. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

A 6.0 m ✕ 8.0 m ✕ 3.0 m room contains air at 20℃. What is the room's thermal energy?

Verified Solution

Key Concepts

Thermal Energy

Specific Heat Capacity

Density of Air