If two liquids of same masses but densities `p_(1)` and `p_(2)`, respectively are mixed, then the density of mixture is given by

Options

A. `p=(p_(1)+p_(2))/(2)`
B. `p=(p_(1)+p_(2))/(2p_(1)p_(2))`
C. `p=(2p_(1)p_(2))/(p_(1)+p_(2))`
D. `p=(p_(1)p_(2))/(p_(1)+p_(2))`

Answer

To find the density of a mixture of two liquids with the same mass but different densities \( p_1 \) and \( p_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Mass of Each Liquid**:
Let the mass of each liquid be \( m \).

2. **Calculate the Volume of Each Liquid**:
The volume of the first liquid (with density \( p_1 \)) can be calculated using the formula:
\[ V_1 = \frac{m}{p_1} \]
The volume of the second liquid (with density \( p_2 \)) is:
\[ V_2 = \frac{m}{p_2} \]

3. **Calculate the Total Volume of the Mixture**:
When the two liquids are mixed, the total volume \( V \) of the mixture is the sum of the individual volumes:
\[ V = V_1 + V_2 = \frac{m}{p_1} + \frac{m}{p_2} \]

4. **Express the Total Volume in a Single Fraction**:
To combine the volumes into a single fraction, we can find a common denominator:
\[ V = m \left( \frac{1}{p_1} + \frac{1}{p_2} \right) = m \left( \frac{p_2 + p_1}{p_1 p_2} \right) = \frac{m (p_1 + p_2)}{p_1 p_2} \]

5. **Calculate the Density of the Mixture**:
The density \( p \) of the mixture can be defined as the total mass divided by the total volume:
\[ p = \frac{\text{Total Mass}}{\text{Total Volume}} = \frac{2m}{V} \]
Substituting the expression for \( V \):
\[ p = \frac{2m}{\frac{m (p_1 + p_2)}{p_1 p_2}} = \frac{2m \cdot p_1 p_2}{m (p_1 + p_2)} \]
Simplifying this gives:
\[ p = \frac{2p_1 p_2}{p_1 + p_2} \]

### Final Result:
The density of the mixture is given by:
\[ p = \frac{2p_1 p_2}{p_1 + p_2} \]