A piece of brass (Cu and Zn) weighs `12.9g` in air. When completely immersed in water, it weighs `11.3g`. Then relative densities of Cu and Zn are `8.9 and 7.1` respectively. The mass of copper in the alloy is

Options

A. `4.6g`
B. 5.6g
C. 7.6 g
D. 8.6g

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a piece of brass weighing 12.9 g in air and 11.3 g when immersed in water. We need to find the mass of copper in the brass alloy, which consists of copper (Cu) and zinc (Zn).

### Step 2: Calculate the Loss of Weight in Water
The loss of weight when the brass is immersed in water can be calculated as follows:
\[ \text{Loss of weight} = \text{Weight in air} - \text{Weight in water} = 12.9 \, \text{g} - 11.3 \, \text{g} = 1.6 \, \text{g} \]
This loss of weight is equal to the weight of the water displaced by the brass.

### Step 3: Relate the Loss of Weight to Volume
The weight of the water displaced can be expressed in terms of volume:
\[ \text{Weight of water displaced} = \text{Volume of water displaced} \times \text{Density of water} \]
Since the density of water is approximately \(1 \, \text{g/cm}^3\), we have:
\[ \text{Volume of water displaced} = 1.6 \, \text{cm}^3 \]

### Step 4: Express Volume in Terms of Mass of Copper and Zinc
Let \(M_1\) be the mass of copper and \(M_2\) be the mass of zinc in the alloy. We know:
\[ M_1 + M_2 = 12.9 \, \text{g} \]
The volume of the brass can also be expressed as:
\[ \text{Volume} = \frac{M_1}{\text{Density of Cu}} + \frac{M_2}{\text{Density of Zn}} = \frac{M_1}{8.9} + \frac{M_2}{7.1} \]
Setting this equal to the volume of water displaced:
\[ \frac{M_1}{8.9} + \frac{M_2}{7.1} = 1.6 \]

### Step 5: Substitute \(M_2\) in Terms of \(M_1\)
From the first equation, we can express \(M_2\):
\[ M_2 = 12.9 - M_1 \]
Substituting this into the volume equation:
\[ \frac{M_1}{8.9} + \frac{12.9 - M_1}{7.1} = 1.6 \]

### Step 6: Solve the Equation
Now we will solve for \(M_1\):
\[ \frac{M_1}{8.9} + \frac{12.9 - M_1}{7.1} = 1.6 \]
Multiplying through by \(8.9 \times 7.1\) to eliminate the denominators:
\[ 7.1M_1 + 8.9(12.9 - M_1) = 1.6 \times 8.9 \times 7.1 \]
Expanding this gives:
\[ 7.1M_1 + 114.801 - 8.9M_1 = 1.6 \times 63.19 \]
Calculating \(1.6 \times 63.19\):
\[ 1.6 \times 63.19 = 101.104 \]
So, we have:
\[ 7.1M_1 - 8.9M_1 + 114.801 = 101.104 \]
Combining like terms:
\[ -1.8M_1 + 114.801 = 101.104 \]
Rearranging gives:
\[ -1.8M_1 = 101.104 - 114.801 \]
Calculating the right side:
\[ -1.8M_1 = -13.697 \]
Dividing by -1.8:
\[ M_1 = \frac{13.697}{1.8} \approx 7.6 \, \text{g} \]

### Conclusion
The mass of copper in the alloy is approximately \(7.6 \, \text{g}\).