If equal masses of two liquids of densities `d_(1) and d_(2)` are mixed together, the density of the mixture is

Options

A. `(d_(1)d_(2))/((d_(1)+d_(2)))`
B. `(2d_(1)d_(2))/((d_(1)+d_(2)))`
C. `(d_(1)d_(2))/(2(d_(1)+d_(2)))`
D. `((d_(1)+d_(2)))/(2)`

Answer

To find the density of a mixture of two liquids with equal masses and different densities \( d_1 \) and \( d_2 \), we can follow these steps:

### Step 1: Define the masses and densities
Let the mass of each liquid be \( m \). The densities of the two liquids are given as:
- Density of liquid 1: \( d_1 \)
- Density of liquid 2: \( d_2 \)

### Step 2: Calculate the volumes of the liquids
Using the formula for density, we can express the volumes of the two liquids in terms of their masses and densities:
- Volume of liquid 1, \( V_1 = \frac{m}{d_1} \)
- Volume of liquid 2, \( V_2 = \frac{m}{d_2} \)

### Step 3: Calculate the total mass and total volume of the mixture
The total mass of the mixture is the sum of the masses of the two liquids:
- Total mass, \( M_{total} = m + m = 2m \)

The total volume of the mixture is the sum of the volumes of the two liquids:
- Total volume, \( V_{total} = V_1 + V_2 = \frac{m}{d_1} + \frac{m}{d_2} \)

### Step 4: Express the total volume in a single fraction
To combine the volumes, we can find a common denominator:
\[ V_{total} = \frac{m}{d_1} + \frac{m}{d_2} = m \left( \frac{1}{d_1} + \frac{1}{d_2} \right) = m \left( \frac{d_1 + d_2}{d_1 d_2} \right) \]

### Step 5: Calculate the density of the mixture
The density of the mixture \( d \) can be calculated using the formula:
\[ d = \frac{M_{total}}{V_{total}} = \frac{2m}{V_{total}} \]
Substituting the expression for \( V_{total} \):
\[ d = \frac{2m}{m \left( \frac{1}{d_1} + \frac{1}{d_2} \right)} = \frac{2}{\frac{1}{d_1} + \frac{1}{d_2}} \]

### Step 6: Simplify the expression
We can simplify this further:
\[ d = \frac{2}{\frac{d_1 + d_2}{d_1 d_2}} = \frac{2d_1 d_2}{d_1 + d_2} \]

### Final Result
Thus, the density of the mixture of the two liquids is:
\[ d = \frac{2d_1 d_2}{d_1 + d_2} \]

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