To determine the time required to heat 20 kg of water from 20°C to 100°C using a 5000 W heater, we need to follow a step-by-step approach:
### Step 1: Understand the Problem and Given Data
- Mass of water ([tex]\(m\)[/tex]): 20 kg
- Initial temperature ([tex]\(T_i\)[/tex]): 20°C
- Final temperature ([tex]\(T_f\)[/tex]): 100°C
- Power of the heater ([tex]\(P\)[/tex]): 5000 W
- Specific heat capacity of water ([tex]\(c\)[/tex]): 4.186 J/g°C (this value is commonly used in such problems)
### Step 2: Convert the Mass into Grams
Since the specific heat capacity is given in J/g°C, we need to convert the mass from kg to grams.
[tex]\[ m = 20 \, \text{kg} \times 1000 \, \frac{\text{g}}{\text{kg}} = 20000 \, \text{g} \][/tex]
### Step 3: Calculate the Temperature Change (ΔT)
[tex]\[
\Delta T = T_f - T_i = 100^\circ\text{C} - 20^\circ\text{C} = 80^\circ\text{C}
\][/tex]
### Step 4: Calculate the Heat Energy Required (Q)
Using the formula [tex]\( Q = mc\Delta T \)[/tex], we calculate the heat energy.
[tex]\[
Q = 20000 \, \text{g} \times 4.186 \, \frac{\text{J}}{\text{g}^\circ\text{C}} \times 80^\circ\text{C}
\][/tex]
[tex]\[
Q = 20000 \times 4.186 \times 80
\][/tex]
[tex]\[
Q = 6697600 \, \text{J}
\][/tex]
### Step 5: Calculate the Time Required
Next, we use the power of the heater to calculate the time. Power (P) is the rate at which energy is transferred, and it's given in watts (Joules per second).
[tex]\[
P = \frac{Q}{t} \implies t = \frac{Q}{P}
\][/tex]
[tex]\[
t = \frac{6697600 \, \text{J}}{5000 \, \text{W}}
\][/tex]
[tex]\[
t = 1339.52 \, \text{s}
\][/tex]
### Step 6: Convert Time into a More Readable Unit (if necessary)
We can convert seconds into minutes for convenience.
[tex]\[
t = \frac{1339.52 \, \text{s}}{60 \, \frac{\text{s}}{\text{min}}} \approx 22.33 \, \text{minutes}
\][/tex]
### Final Answer
It will take approximately 1339.52 seconds (or about 22.33 minutes) to heat 20 kg of water from 20°C to 100°C using a 5000 W heater.
