Density of water is `1.00 g//cm^(3)` and density of ethanol is `0.9 g//cm^(3)`. If these two liquids were mixed to prepare a dilute solution then

Options

A. density of the solution `gt 1.00 g//cm^(3)`
B. density of the solution `lt 1.00 g//cm^(3)`
C. density of the solution `lt 0.9 g//cm^(3)`
D. density of the solution `= 1 g//cm^(3)`

Explanation

To solve the problem of finding the density of a mixture of water and ethanol, we will follow these steps:

### Step 1: Understand the Densities
We know the densities of the two liquids:
- Density of water (ρ_water) = 1.00 g/cm³
- Density of ethanol (ρ_ethanol) = 0.90 g/cm³

### Step 2: Define the Volumes
Assume we take:
- A mL of water
- B mL of ethanol

### Step 3: Calculate Masses
Using the formula for mass (mass = density × volume):
- Mass of water (m_water) = ρ_water × A = 1.00 g/cm³ × A mL = A grams
- Mass of ethanol (m_ethanol) = ρ_ethanol × B = 0.90 g/cm³ × B mL = 0.90B grams

### Step 4: Calculate Total Mass
Total mass (m_total) of the solution is the sum of the masses of water and ethanol:
\[ m_{total} = m_{water} + m_{ethanol} = A + 0.90B \]

### Step 5: Calculate Total Volume
Total volume (V_total) of the solution is the sum of the volumes of water and ethanol:
\[ V_{total} = A + B \]

### Step 6: Calculate Density of the Mixture
Density of the mixture (ρ_mixture) is given by the formula:
\[ ρ_{mixture} = \frac{m_{total}}{V_{total}} = \frac{A + 0.90B}{A + B} \]

### Step 7: Analyze the Result
To analyze the density further, we can express it in terms of A and B:
- The numerator is \( A + 0.90B \)
- The denominator is \( A + B \)

### Step 8: Determine the Range of Density
Since:
- \( A \) is a positive quantity (volume of water)
- \( B \) is a positive quantity (volume of ethanol)

We can conclude:
1. If \( B = 0 \) (only water), then \( ρ_{mixture} = 1.00 \) g/cm³.
2. If \( A = 0 \) (only ethanol), then \( ρ_{mixture} = 0.90 \) g/cm³.

Thus, the density of the mixture will be between 0.90 g/cm³ and 1.00 g/cm³.

### Conclusion
Therefore, the density of the solution when water and ethanol are mixed will be:
\[ 0.90 \, \text{g/cm}^3 < ρ_{mixture} < 1.00 \, \text{g/cm}^3 \]

### Final Answer
The density of the solution is less than 1 g/cm³ and greater than 0.90 g/cm³.

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