An electric generating plant boils water to produce high-pressure steam. The steam spins a turbine that is connected to the generator. a. How many liters of water must be boiled to fill a 5.0 m^3 boiler with 50 atm of steam at 400°C?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice from together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A pressure cooker operates at a pressure of 2.0 atmospheres and a temperature of 120 degrees Celsius. A 1.0 liter bottle is used to collect the steam that is released from the safety valve. What is the amount of water in liters required to produce enough steam to fill the bottle? Assume the steam inside the bottle is at 2. atmospheres and 120 degrees Celsius. OK. So we're given some multiple choice answers here. They're all in the same units of leaders. Let's read them off to see what our final answer might be. A is 1.1 multiplied by 10 to the power of negative three B is 6.2 multiplied by 10 to the power of negative three C is 0.11 and D is 0.62. OK. So first off, let us recall and use the ideal gas law equation which states that pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by the temperature. So, but in this case, to be specific note that N is the number of moles of steam, V is the volume of steam inside the bottle and pressure is the pressure of steam awesome. And then T is the temperature of steam. OK. So we need to rearrange the ideal gas law equation to solve for the numbers of mole of steam. So when we rearrange that to sol for the numbers of moles of steam, we get that N equals the pressure of steam multiplied by the volume of steam inside the bottle, divided by the universal gas constant multiplied by the temperature of steam. So at this stage, we can plug in all of our known variables to solve for the number of moles. So let's do that. So we're given that the pressure is two atmospheres. But in our final answer, we have to get it of the atmosphere is to cancel out. So we need to multiply it by atmospheric pressure. So let's do that. So we have two atmospheres for the pressure and then we need to multiply it by atmospheric pressure which is 1.13 multiplied by 10 to the fifth power pass scales. But the units for the atmosphere pressure is past scales divided by atmospheres. So the atmospheres cancel out and then we multiply it by the volume which we need the volume in cubic meters. A set of liters means we're given given as our volume as 1.0 liters. So all we have to do is just use quick, use some dimensional analysis and solve for that really quick to convert it to meters cube. So in one cubic meter there is liters. OK. So all we have to use, all we have to do is use dimensional analysis. OK. So then the numerical value for the universal gas constant is 8.31 Jews per mole multiplied by Calvin multiplied by the temperature. But let's pause here for a second here because the temperature is given to us in degrees Celsius, we need to convert degrees Celsius two degrees Kelvin. So 100 degrees Celsius, I mean you convert to Kelvin. So all we have to do is just to convert, you know, degrees Celsius to Kelvin is you just take the degrees Celsius and you add 273. So that will equal when you add those two together Kelvin K 393 Kelvin. Awesome. So when you plug that into a calculator, you should get for your number of moles, 0.62 moles. So let's make a quick note here that one mole of water produces one mole of steam and that the molecular mass of water. So it's let's write this down, I'm gonna denote as capital M subscript W So the molecular mass of water is equal to 18 g per mole. So in one mole of water, we have 18 oh sorry, one mole of water will have a mass of 18 g. Ok. So let's note that the mass. So, so we're trying to find the mass of steam that the mass of steam will equal the number of moles of water. So it was 0.62 mole multiplied by 18 g per mole. So the moles will cancel out just leaving grams. And we're given, when we put that plug that into a calculator, 1.11 g, but we need to convert grams to kilograms. So we just need to use dimensional analysis. So in one kg, there is 1000 grams. OK? OK. So we could keep it like that. We don't have to multiply it out into a calculator if you can if you want to. But for simplicity, let's just keep it like that. OK. So then at this point, we need to recall and use the density relationship re equation to solve for the water's volume. So let's recall and remember that the density of water row equals the mass of water divided by the volume of water. So let's rearrange to solve for the volume of water. So V subscript W the volume of water is equal to the mass of water divided by row. And let's make a quick note here that row, the density of water equals kilograms per meter cubed. OK. So let's plug in our known variables to sulfur the volume of water. OK. So we just determine the mass of water to be 1.11 g multiplied by one. So dimensional analysis. So in one kg there is 1000 g. So one kg divided by 1000 g and then divided by the density of water, which was 1000 kg per meter cubed. OK. So when you plug that into a calculator, the volume of water will be 1.11 multiplied by 10, the power of negative six m cubed. But there are final answers if you remember and the multiple choice are all in liters. So we need to use dimensional analysis to convert cubic meters back to liters. So there's 1000 liters and one cubic meter. So when you multiply or plug that into a calculator, I should say when you plug that into a calculator, you should get that. The volume of water in liters is 1.11 multiplied by 10 to the power of negative three liters. And that is our final answer. Hooray, we did it. So our final answer must be the letter A 1.1 multiplied by 10 to the power of negative three liters. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.