To solve the problem of calculating the energy required to raise the temperature of water from \(15^\circ C\) to \(25^\circ C\) in a tank containing \(10^5\) liters of water, we will follow these steps:
### Step 1: Convert the volume of water to mass
1. We know that the density of water is \(1 \, \text{g/cm}^3\) or \(1 \, \text{g/mL}\).
2. Since \(1 \, \text{L} = 1000 \, \text{mL}\), we can convert liters to grams:
\[
10^5 \, \text{L} = 10^5 \times 1000 \, \text{mL} = 10^8 \, \text{g}
\]
### Step 2: Calculate the change in temperature
1. The initial temperature (\(T_i\)) is \(15^\circ C\) and the final temperature (\(T_f\)) is \(25^\circ C\).
2. The change in temperature (\(\Delta T\)) is calculated as:
\[
\Delta T = T_f - T_i = 25^\circ C - 15^\circ C = 10^\circ C
\]
### Step 3: Use the formula for heat energy
1. The formula to calculate the heat energy (\(Q\)) required to change the temperature of a substance is:
\[
Q = m \cdot s \cdot \Delta T
\]
where:
- \(m\) = mass of the water in grams
- \(s\) = specific heat capacity of water = \(4 \, \text{J/(g} \cdot \text{°C)}\)
- \(\Delta T\) = change in temperature in °C
### Step 4: Substitute the values into the formula
1. Substitute \(m = 10^8 \, \text{g}\), \(s = 4 \, \text{J/(g} \cdot \text{°C)}\), and \(\Delta T = 10 \, \text{°C}\):
\[
Q = 10^8 \, \text{g} \cdot 4 \, \text{J/(g} \cdot \text{°C)} \cdot 10 \, \text{°C}
\]
\[
Q = 4 \times 10^9 \, \text{J}
\]
### Step 5: Convert joules to kilojoules
1. Since \(1 \, \text{kJ} = 1000 \, \text{J}\), we convert the energy from joules to kilojoules:
\[
Q = \frac{4 \times 10^9 \, \text{J}}{1000} = 4 \times 10^6 \, \text{kJ}
\]
### Step 6: Identify the value of \(n\)
1. The problem states that the energy required is \(n \times 10^6 \, \text{kJ}\).
2. From our calculation, we have:
\[
4 \times 10^6 \, \text{kJ} = n \times 10^6 \, \text{kJ}
\]
Thus, \(n = 4\).
### Final Answer
The value of \(n\) is \(4\).
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