An immersion heater must increase the temperature of 1.50 kg of water from 10.0°C to 50.0 °C in 10.0 minutes while operating at 110V. If the specific heat capacity of water is 4186 J/kg/°C, calculate the required resistance of the heater.
To calculate the required resistance of the immersion heater, we need to use the formula:
P = IV
Where P is the power, I is the current, and V is the voltage. We can rearrange this formula to solve for I:
I = P / V
Next, we can calculate the power by using the formula:
Q = mcΔT
Where Q is the energy transferred, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature. Rearranging this formula gives us:
P = Q / t
Where t is the time taken. Substituting the values, we get:
P = (m * c * ΔT) / t
Now we have the power, we can substitute it back into the initial formula to solve for the current:
I = [(m * c * ΔT) / t] / V
Finally, we can use Ohm's law to calculate the resistance. Ohm's law states that:
V = IR
Rearranging this formula gives us:
R = V / I
Substituting the values, we get:
R = V / [(m * c * ΔT) / t]
Now, let's calculate the values:
m = 1.50 kg (mass of water)
c = 4186 J/kg/°C (specific heat capacity of water)
ΔT = (50.0 °C - 10.0 °C) = 40.0 °C (change in temperature)
t = 10.0 minutes = 600 seconds (time taken)
V = 110V (voltage)
Substituting these values into the formula, we have:
R = 110 / [(1.50 * 4186 * 40.0) / 600]
Simplifying further, we get:
R = 110 / [(1.50 * 4186 * 40.0) / 600]
R = 110 / [251160 / 600]
R = 110 / 418.6
R ≈ 0.26 Ω
Therefore, the required resistance of the immersion heater is approximately 0.26 ohms.