Question

How long would a 5.00 kW space heater have to run to put into a kitchen the same amount of heat as a refrigerator (coefficient of performance = 2.00) does when it freezes 2.00 kg of water at 26.0°C into ice at 0.0°C?

Answer 1

A 5.00 kW space heater would need to run for approximately 43.5 seconds to produce the same amount of heat as a refrigerator that freezes 2.00 kg of water at 26.0°C into ice at 0.0°C.

Step-by-step explanation:

To determine how long a 5.00 kW space heater would need to run to produce the same amount of heat as a refrigerator that freezes 2.00 kg of water at 26.0°C into ice at 0.0°C, we need to compare the energy required for each process.

The energy required to freeze the water can be calculated using the formula:

Energy = mass * specific heat capacity * temperature change

For freezing 2.00 kg of water, the energy required is:

Energy = 2.00 kg * 4,180 J/kg°C * (26.0°C - 0.0°C) = 217,680 J

Now, let's calculate the time it would take for the space heater to produce the same amount of heat:

Time = Energy / Power

Time = 217,680 J / 5,000 W = 43.5 seconds

Therefore, the 5.00 kW space heater would need to run for approximately 43.5 seconds to produce the same amount of heat as the refrigerator.

Answer 2

The corrected time required for the 5.00 kW space heater to generate the same amount of heat as removed from the water is approximately 43.58 seconds.

Of course! Let's go through the problem step by step to find the time required for the space heater to generate the same amount of heat as removed from the water.

`\(\Delta T = 0.0^(\circ)C - 26.0^(\circ)C = -26.0^(\circ)C\)`

Given:

- Mass of water,
`\(m = 2.00 \, \text{kg}\)`

- Specific heat capacity of water,
`\(c = 4186 ^(\circ)\, \text{J/kg C}\)`

- Temperature change,

- Heat removed from the water,
`\(Q = mc\Delta T\)`

1. Calculate the heat removed from the water:

`\[Q = mc\Delta T = 2.00 \, \text{kg} * 4186^(\circ) \, \text{J/kg C} * (-26.0^(\circ)C)\]`

`\[Q = -217,912 \, \text{J}\]` (negative because heat is being removed)

2. Now, find the time the 5.00 kW space heater needs to run to generate the same amount of heat:

`\[ \text{Time} = \frac{\text{Energy removed from water}}{\text{Power of space heater}} \]`

Given that the space heater's power is \(5.00 \, \text{kW} = 5.00 \times 10^3 \, \text{W}\):

`\[ \text{Time} = \frac{-217,912 \, \text{J}}{5.00 * 10^3 \, \text{W}} \]`

Calculating the time:

`\[ \text{Time} = \frac{-217,912 \, \text{J}}{5.00 * 10^3 \, \text{W}} \approx 43.58 \, \text{s} \]`