An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 m>s when the gas temperature is 22.0°C. For a certain experiment, you need to have the same oscillator produce sound of wavelength 28.5 cm in this gas. What should the gas temperature be to achieve this wavelength?

Video transcript

Hey, everyone in this problem, we're told that when the atmosphere temperature is zero degrees Celsius, a vibrating rod mounted on a hot air balloon generates longitudinal waves traveling at a speed of 331 m per second. The vi vibrating rod's frequency is 1000 Hertz. The hot air balloon rises in the sky at a height where the temperature is t the waves have a wavelength of 0.28 m to obtain this wavelength. What should be the atmospheric temperature? We're told to consider air as an ideal gas. We're given four answer choices. Option A T is equal to negative 100.7 degrees Celsius. Option BT is equal to negative 77.6 degrees Celsius. Option C T is equal to negative 46.5 degrees Celsius. And option D T is equal to negative 25.5 degrees Celsius. Now, what we're trying to do here is find a temperature at a particular wavelength. OK. So let's recall, we have the following equation we have that the speed of V is equal to the square root of gamma RT divided by M K where gamma is a ratio of heat capacities R is the gas constant T is the temperature and M is the molar mass. Now, you might be saying, OK, that has temperature but it has speed and not wavelength. But recall that we can relate the speed and the wavelength through the following the frequency is equal to the speed V divided by the wavelength lambda. Now, we're given both the frequency and the wavelength. And so we can find that speed and use this equation. Now, if we do that, we'll have a V, we know the gas constant R but we don't know gamma or M in this situation. Hm. So we won't be able to find the temperature we're looking for yet. So let's first look at the situation we were given initially where we start at zero degrees Celsius, we're given the temperature and we're given a speed. So let's use that to find gamma and M and then we can use those values to find the temperature we're looking for. So we're gonna use the exact same equation B is equal to the square root of gamma RT divided by. Now V is our speed 331 m per second, that's equal to the square root of gamma multiplied by the gas constant R just approximately 8. jewels per no. Kelvin not supplied by their temperature. Now, in this case, we're looking when we have a temperature T of zero degrees Celsius in order to convert this to Kelvin, we wanna add 273 OK? We're at zero degrees Celsius. And so this is gonna be equal to 273 Kelvin. OK. And that's important in our equation, we want to be using the absolute temperature, OK. If we were to use the temperature in Celsius, we would have zero on the numerator, the entire right hand side would go to zero. And so this equation would not be true. OK. So we need to use that absolute temperature in Calvin. And so we have 273 Calvin in the numerator inside that square root. And then we divide all of that by M. Now, our unit of Calvin, that's gonna divide it. OK? We have per Calvin on our gas constant. We have Calvin on our temperature. Those divide up, we're gonna square both sides and we're trying to isolate gamma and death. We get 109,000 561. We have meters per second when we square that we get meters squared per second squared. On the right hand side, we have gamma multiplied by, by 8.3 Jews per mole multiplied by 273 two divided by M. Now, you might look at this equation and go how are we supposed to solve for gamma and M and we have one equation with two unknown values and that's true. We wouldn't be able to solve for each of them on their own. However, when we use them in the next equation, it's the same equation. So all we need to know is the ratio gamma divided by M. We don't need to know each of them on their own. OK. So we're gonna solve for gamma divided by M gamma divided by M is going to be equal to 109,561 m squared per second squared, divided by 8. Jews per mole multiplied by 273. OK? And this is gonna give us gamma divided by M is equal to 0.3521 in our unit. Here, it's gonna be meters squared more divided by second squared job. All right. So it's a little bit of a messy unit, but it's because we're looking at the ratio of these two values and it will all work out when we plug it into the next equation. You'll see that in just a minute. OK. So we have G M divided by M. We're trying to find the temperature. We know the gas constant R. Now, the last thing we have to do before we can use this equation to find that temperature is to find V. OK. So I'm gonna go back up, remember we mentioned that the frequency is related to the speed and the wavelength. So we have a way to calculate that speed. The speed V is going to be equal to the frequency at multiplied by the wavelength we were given both. And so we can substitute those undefined. We have the frequency is 1000 Hertz multiplied by the wavelength of 0.28 m. And we get a speed of 280 m per second. So now we have it all, we have our speed, we have the ratio of gamma divided by M. We have our gas constant R. The only unknown that's left is that temperature T that we're looking for. Let's give ourselves some more space at the bottom here and work this out. We're starting with the same equation V is equal to the square root of gamma RT divided by M. Our speed is 280 m per second, which is equal to the square root of gamma. 48. meter squared mole divided by second squared jewel multiplied by the gas constant 8. jewels per mole. Kelvin multiplied by T that temperature we're interested in. And remember this first value 48.3521 is gamma divided by M. OK. So we don't need to divide anything else. That's the entire term. Now we look at some of these units and they will divide it. We have a mole and then we have a per mole, we have a jewel and a per jewel and those will divide it So inside the square root, we're left with meters squared per second squared. And then we're left with eight per Kelvin. We're trying to solve for T let's square both sides. On the left hand side, we get 78,400 m squared per second squared. And on the right hand side, when we multiply our two terms together, we get 401 0.322, 43, the units we are left with is meter squared per second squared. Kelvin multiplied by T we're gonna divide both sides by this 401.32243 to isolate for the temperature. When we do that, the unit of meter squared per second squared is gonna divide out. We're gonna be left with the unit of Calvin and the numerator, which is exactly what we want for temperature. And so we get that this temperature is equal to 195.354. Kelvin. Now, if we look at our answer choices, the answer choices we were given were all in degrees Celsius. So to convert back to degrees Celsius, we need to subtract 273. Kelvin. They have 195. minus 273 which will give us negative 0.646 degrees Celsius, which is the final answer we were looking for. That is the temperature when we have the wavelength of 0.28 m. If we go back up to our answer traces, we're gonna round this to a single decimal place and we see that our answer corresponds with answer choice. BT is equal to negative 77.6 degrees Celsius. Thanks everyone for watching. I hope this video helped see you in the next one.