A heat engine is found to have an
efficiency of 0.620. The engine performs 300 joules of
work per cycle. How much is the engine heated per
cycle? How much heat is rejected by the
engine per cycle?
Okay, so first of all, let’s have a
look at the bit of information that tells us that the efficiency of the engine is
0.620. Let’s call the efficiency of the
engine 𝜂, and we’ve been told that this is equal to 0.620. Now we can recall that efficiency
is defined as the useful energy output from the engine divided by the total energy
input to the engine.
Now for a heat engine, the useful
energy output is the work performed by the engine. And we know that, in this case, 300
joules of work are performed by the engine per cycle. And so we can replace the useful
energy output with 300 joules per cycle. And of course, we know from earlier
that 𝜂 is 0.620.
But then because we’ve got 300
joules per cycle, in order for this equation to make sense, we can’t actually just
have the total energy input. What we actually need in the
denominator is the total energy input per cycle because, this way, we can think of
it as the per-cycle bits canceling out, which leaves us with a useful energy output
divided by the total energy input, exactly as we had earlier.
Anyway, so we can rearrange the
equation to solve for the total energy input per cycle, which ends up being 300
joules per cycle divided by 0.620. This simplifies to 484 joules per
cycle, at which point we can also remember that the energy input for a heat engine
is actually heat. So what we’ve just found out is how
much the engine is heated per cycle.
Now since the question already says
per cycle, we don’t need to include that in our answer because the engine is heated
by 484 joules per cycle. And so we can move on to the next
part. We need to work out how much heat
is rejected by the engine per cycle.
To answer this, we need to recall
that, for an engine, some of the heat that goes into the engine is converted into
work, which is the output, and the rest of that heat is rejected. Now we know that the heat into the
engine is 484 joules, per cycle anyway, and we also know that the work done by the
engine is 300 joules per cycle. So whatever is left of that heat is
going to be the rejected heat. In other words, we can say that 484
joules minus 300 joules is the heat rejected per cycle. And this ends up being 184 joules,
at which one we found the heat rejected by the engine per cycle. So that is our final answer.