Steam locomotives have an efficiency of 17.0% and operate with a hot steam temperature of $425^{\circ }\text{C}$. (a) What would the cold reservoir temperature be if this were a Carnot engine? (b) What would the maximum efficiency of this steam engine be if its cold reservoir temperature were $150^{\circ }\text{C}$?

Video Transcript

This is College Physics Answers with Shaun Dychko. A steam engine has hot steam at a temperature of 425 degrees Celsius and its efficiency is 0.170. So, we're asked to find out, what would the cold reservoir temperature be if this was the Carnot efficiency? So, Carnot efficiency is 1 minus the cold reservoir temperature divided by the hot and both of these have to be expressed in Kelvin, they are absolute temperatures. And, we'll solve this for Tc eventually. First, we'll add Tc over Th to both sides and then we'll subtract the Carnot efficiency from both sides. And so, we have Tc over Th equals 1 minus efficiency. And, then we multiply both sides by the hot reservoir temperature and we solve for the cold reservoir temperature. So, it will be the hot reservoir temperature times 1 minus efficiency, so that's 425 degrees Celsius converted into Kelvin by adding 273.15 and then multiplied by 1 minus 0.170. This works out to this temperature in Kelvin, but it's nice to write it in Celsius since we have a better intuitive understanding of that unit. So, we'll subtract 273.15 from it and get 306 degrees Celsius. Now, if the cold reservoir temperature was actually 150 degrees Celsius, the question is, what would the maximum possible efficiency be, which is, in other words, the Carnot efficiency? So, that's 1 minus cold temperature divided by hot temperature, so that is 1 minus cold temperature in Kelvin divided by the steam temperature in Kelvin. This works out to 0.394.