A Carnot refrigerator is used in a kitchen in which the temperature is kept at 301 K. This refrigerator uses 216 J of work to remove 2360 J of heat from the food inside. What is the temperature inside the refrigerator?
η=W/Q =(T₁-T₂)/T₁=1-(T₂/T₁),
T₂=T₁(1 – W/Q)= 301(1- 216/2360)=280.7 K
To find the temperature inside the refrigerator, we can use the Carnot refrigerator efficiency formula:
Efficiency = 1 - (Tc/Th)
Where:
Efficiency is the ratio of the useful work output to the input heat
Tc is the temperature of the cold reservoir (inside the refrigerator)
Th is the temperature of the hot reservoir (kitchen temperature)
First, let's find the efficiency of the refrigerator:
Efficiency = (work input - heat removed) / work input
Efficiency = (216 J - 2360 J) / 216 J
Efficiency = -2144 J / 216 J
Efficiency = -9.93
The efficiency cannot be negative, so there must be a calculation or sign error. Please check your values again and ensure the correct signs are used in the formula.
To find the temperature inside the refrigerator, we can use the equation for the coefficient of performance (COP) of a Carnot refrigerator:
COP = Q_cold / W,
where COP is the coefficient of performance, Q_cold is the amount of heat removed from the cold reservoir (food), and W is the work done by the refrigerator.
In this case, we are given Q_cold = 2360 J and W = 216 J.
Thus, we can rearrange the equation to solve for the temperature:
COP = Q_cold / W,
COP = 2360 J / 216 J.
Simplifying, we find:
COP = 10.93.
Now, we need to find the temperature inside the refrigerator. The COP for a Carnot refrigerator is given by:
COP = T_cold / (T_hot - T_cold),
where T_cold is the temperature inside the refrigerator (in Kelvin) and T_hot is the temperature of the hot reservoir (kitchen temperature) in Kelvin.
We are given T_hot = 301 K.
Substituting the values into the equation, we have:
10.93 = T_cold / (301 K - T_cold).
To solve for T_cold, we can rearrange the equation as:
10.93 * (301 K - T_cold) = T_cold.
Expanding the equation and rearranging, we get:
3291.93 K - 10.93 * T_cold = T_cold.
Simplifying, we find:
3291.93 K = 11.93 * T_cold.
Finally, we can solve for T_cold:
T_cold = 3291.93 K / 11.93.
Calculating this expression, we find:
T_cold ≈ 275.81 K.
Therefore, the temperature inside the refrigerator is approximately 275.81 Kelvin.