A 750 g aluminum pan is removed from the stove and plunged into a sink filled with 10.0 L of water at 20.0°C . The water temperature quickly rises to 24.0°C. What was the initial temperature of the pan in °C and in °F?

Video transcript

Hey, everyone. So this problem is dealing with conservation of energy and heat transfer. Let's see what it's asking us. A Black Smith places a 390 g iron horseshoe inside a tank that contains five liters of water at 298 degrees. Kelvin. A thermometer shows an increase to the water temperature reaching 301 degrees. Kelvin determine the original temperature in degrees Celsius in Fahrenheit of the horseshoe before it was immersed in the water. So our multiple choice answers here are a degrees Celsius, 663 degrees Fahrenheit B 386 degrees Celsius, 727 degrees Fahrenheit, C 420 degrees Celsius, 724 degrees Fahrenheit or D 420 degrees Celsius, 788 degrees Fahrenheit. So we're going to solve this problem to find the original temperature in Kelvin and then we'll do some unit conversion to get the answers in Celsius in Fahrenheit. And so the first thing we can do is recall that we're in a closed system. So we have conservation of energy. So the heat of the horseshoe plus the heat of the water is going to equal zero. And this heat energy is given by the mass multiplied by the specific heat capacity multiplied by the change in temperature. And so we'll have for the Horseshoe MC. Delta T is equal to negative, the same for the water mass of the water multiplied by the specific heat capacity of the water multiplied by the changing temperature of the water. So we're looking for the original temperature of the horseshoe. So that's going to be contained in this delta T horseshoe term. So delta T is T horseshoe final minus T horseshoe initial and then that equals negative mass of the water. Specific heat capacity of the water. Delta T of the water is going to be t water final minus T water initial, all divided by the mass of the horseshoe multiplied by the specific heat capacity of the horseshoe. Now the final temperature of the water and the final temperature of the horseshoe are the same. So they reach that equilibrium temperature of 300 and one degrees Kelvin. So we'll just call that t final. And then looking at the other terms here, we're solving for our horseshoe temperature, initial temperature of the horseshoe, the mass of the water was given was not given to us, the volume of the water was. So we can solve for that where we have five liters. And so we can recall that a liter of water equals a kilogram. And so that's just five kg. Our specific heat capacity of water, it's a constant. So we can recall that's 4, 4186 Jews per kilogram k. Our initial heat or it's our initial temperature of the water is 298 k. And then the mass of the horseshoe was given as 390 g. So we're going to rewrite that as 3900.39 kg to keep us in standard units. And then the specific heat capacity of iron which the horse she is made of, you can look up is 450 Jews per kilogram K. And so now we have everything we need to solve for our initial temperature of the horseshoe in tell them and then you will convert to Celsius and be so we have negative five kg multiplied by jewels per kilogram K multiplied by 301 K minus 298 K all divided by 0.39 kg multiplied by Jews per kilogram K and then minus 301. This is going to be a negative. And so we'll divide everything by negative one, get our final answer, plug this all in and we get 659 Kelvin. And so now we just need to recall how to convert Kelvin to Celsius which is simply adding, sorry, simply subtracting 700. And excuse me, Kelvin to Celsius is the temperature Kel Kelvin minus 273 degrees. Kelvin. So we have 659 Kelvin minus 273 degrees equals 386 degrees Celsius. That's the answer in Celsius. And then Celsius to Fahrenheit is our temperature in Celsius multiplied by 9/5 in that quantity plus 32 degrees. And so that gives us degrees Fahrenheit. And so when we look at our multiple choice answers that aligns with answer choice B so B is the correct answer for this problem. So that's all we have for this one. We'll see you in the next video.