Heat engine X takes four times more energy by heat from the hot reservoir than heat engine Y. Engine X delivers two times more work, and it reject seven times more energy by heat to the cold reservoir than heat engine Y. Find the efficiency of heat engine X and heat engine Y.
Qin,x - Qout,x = Work,x
Qin,y - Qout,y = Work,y
Work,x = 2 Work,y
Qin,x = 4 Qin,y
Qout,x = 7 Qout,y
4Qin,y - 7Qout,y = 2 Work,y
7Qin,y - 7Qout,y = 7 Work,y
3Qin,y = 5Work,y
Since Efficiency = W/Qin,
Efficiency,y = 3/5
Work,x/Qin,x = 2Work,y/4Qin,y
Efficiency,x = (1/2)Efficiency,7 = 3/10
I got everything up until i got to the
7Qin,y - 7Qout,y= 7 Work,y
where did those numbers come from?
It is the same as the second equation on the list, with both sides multiplied by 7. It was one step in elimimating the Qout,y variable when combined with the equation above it.
eat engine X takes in four times more energy by heat from the hot reservoir seven times more energy by heat to the cold reservoir than heat engine Y find the efficiency of Heat engine X and heat engine Y
Ah, I see what you're getting at! It's all about manipulating equations to get rid of variables. In this case, we multiplied the second equation by 7 to make the coefficients of Qout,y on both sides match up. This allows us to subtract the two equations and eliminate Qout,y from the equation!
It's like doing math gymnastics, but with numbers instead of somersaults. Quite a workout for the brain, if I do say so myself!
Let's break down the steps to understand how those numbers were obtained.
Given:
1. Engine X takes four times more energy by heat from the hot reservoir than Engine Y.
Qin,x = 4Qin,y
2. Engine X delivers two times more work than Engine Y.
Work,x = 2Work,y
3. Engine X rejects seven times more energy by heat to the cold reservoir than Engine Y.
Qout,x = 7Qout,y
Now, let's find the values of Qin,x, Qin,y, Qout,x, and Qout,y in terms of Work,y.
From equation 2:
Work,x = 2Work,y
From equation 1:
Qin,x = 4Qin,y
Using the definition of efficiency (Efficiency = Work / Qin), we can rewrite equation 2 as:
Efficiency,x * Qin,x = 2Efficiency,y * Qin,y
Substituting from equation 1:
Efficiency,x * 4Qin,y = 2Efficiency,y * Qin,y
Simplifying:
4Efficiency,x = 2Efficiency,y
Efficiency,x = Efficiency,y / 2
Now, let's consider Qout,x and Qout,y.
From equation 3:
Qout,x = 7Qout,y
Again, using the definition of efficiency, we can rewrite equation 3 as:
Qout,x = 7Efficiency,y * Work,y
Substituting from equation 2:
7Efficiency,y * Work,y = 7Qout,y
Simplifying:
Efficiency,y = Work,y / Qout,y
Now, let's substitute this value of Efficiency,y back into the equation we found for Efficiency,x:
Efficiency,x = (Work,y / Qout,y) / 2
Simplifying:
Efficiency,x = Work,y / (2 * Qout,y)
So, the final expression for the efficiency of Engine X (Efficiency,x) in terms of Work,y and Qout,y is Efficiency,x = Work,y / (2 * Qout,y).
I apologize for any confusion caused by the initial calculation steps.
To understand where the numbers in the equation 7Qin,y - 7Qout,y = 7 Work,y came from, let's break it down step by step.
We know that Qin,x - Qout,x = Work,x (equation 1)
And we know that Qin,y - Qout,y = Work,y (equation 2)
Since Work,x = 2 Work,y, we can substitute this in equation 1:
Qin,x - Qout,x = 2 Work,y
Now, we also know that Qin,x = 4 Qin,y (given), so we can replace Qin,x in the equation above:
4 Qin,y - Qout,x = 2 Work,y
And we know that Qout,x = 7 Qout,y (given), so we can replace Qout,x in the equation:
4 Qin,y - 7 Qout,y = 2 Work,y
To simplify this equation, we can multiply both sides by 7:
28 Qin,y - 49 Qout,y = 14 Work,y
And dividing both sides by 7 gives us:
4 Qin,y - 7 Qout,y = 2 Work,y
This is the same equation as mentioned before, which is used to eliminate the Qout,y variable when combined with the equation above it.
So, 7Qin,y - 7Qout,y = 7 Work,y is derived from the given information and substitution in equations 1 and 2.