Calculate the resulting temperature when 20 g of boiling water is poured into an ice-cold brass vessel `("specific heat = 0.1 cal g"^(-1) .^(@)C^(-1))` of mass 100 g

Options

A. `66.66.^(@)C`
B. `6.66.^(@)C`
C. `0.66.^(@)C`
D. `50.^(@)C`

Answer

To solve the problem of calculating the resulting temperature when 20 g of boiling water is poured into an ice-cold brass vessel, we can follow these steps:

### Step 1: Define the variables
Let:
- \( m_w = 20 \, \text{g} \) (mass of boiling water)
- \( m_b = 100 \, \text{g} \) (mass of brass vessel)
- \( c_b = 0.1 \, \text{cal/g°C} \) (specific heat of brass)
- \( c_w = 1 \, \text{cal/g°C} \) (specific heat of water)
- \( T_f \) = final temperature (the temperature we need to find)

### Step 2: Set up the heat transfer equation
The heat lost by the boiling water will be equal to the heat gained by the brass vessel. Therefore, we can write the equation as:
\[ \text{Heat lost by water} = \text{Heat gained by brass} \]
This can be expressed mathematically as:
\[ m_w \cdot c_w \cdot (T_i - T_f) = m_b \cdot c_b \cdot (T_f - T_b) \]
Where:
- \( T_i = 100°C \) (initial temperature of boiling water)
- \( T_b = 0°C \) (initial temperature of brass vessel)

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation gives:
\[ 20 \cdot 1 \cdot (100 - T_f) = 100 \cdot 0.1 \cdot (T_f - 0) \]
This simplifies to:
\[ 20(100 - T_f) = 10T_f \]

### Step 4: Expand and rearrange the equation
Expanding the left side:
\[ 2000 - 20T_f = 10T_f \]
Now, rearranging the equation to isolate \( T_f \):
\[ 2000 = 20T_f + 10T_f \]
\[ 2000 = 30T_f \]

### Step 5: Solve for \( T_f \)
Now, divide both sides by 30:
\[ T_f = \frac{2000}{30} = 66.67°C \]

### Final Answer
The resulting temperature when 20 g of boiling water is poured into an ice-cold brass vessel is approximately:
\[ \boxed{66.67°C} \]