The ratio of specific heats of two spheres is 2:3, radii 1:2, emissivity 3:1 and density 1:1. Initially, they are kept at same temperatures in atmosphere. Which sphere will cool at a faster rate initially.

Explanation

To determine which sphere will cool at a faster rate initially, we will analyze the given parameters and apply the formula for the cooling rate.

### Step-by-Step Solution:

1. **Identify the given ratios:**
- Ratio of specific heats (C1:C2) = 2:3
- Ratio of radii (r1:r2) = 1:2
- Ratio of emissivity (E1:E2) = 3:1
- Ratio of density (ρ1:ρ2) = 1:1

2. **Understand the formula for cooling rate:**
The cooling rate can be expressed as:
\[ \text{Cooling Rate} \propto \frac{E \cdot A}{m \cdot C} \cdot (T^4 - T_0^4) \]
Since both spheres are initially at the same temperature, the term \((T^4 - T_0^4)\) will cancel out when comparing the two spheres. Thus, we need to focus on the term \(\frac{E \cdot A}{m \cdot C}\).

3. **Calculate the area (A) for both spheres:**
The surface area of a sphere is given by:
\[ A = 4\pi r^2 \]
For the ratio of areas:
\[ \frac{A_1}{A_2} = \frac{4\pi (r_1^2)}{4\pi (r_2^2)} = \frac{r_1^2}{r_2^2} \]
Given \(r_1:r_2 = 1:2\), we find:
\[ \frac{A_1}{A_2} = \frac{1^2}{2^2} = \frac{1}{4} \]

4. **Calculate the mass (m) for both spheres:**
The mass of a sphere can be calculated using:
\[ m = \rho \cdot V = \rho \cdot \frac{4}{3}\pi r^3 \]
Since the densities are equal, we can compare the volumes:
\[ \frac{m_1}{m_2} = \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]
Given \(r_1:r_2 = 1:2\), we find:
\[ \frac{m_1}{m_2} = \frac{1^3}{2^3} = \frac{1}{8} \]

5. **Substitute the values into the cooling rate ratio:**
Now we can substitute the values into the cooling rate ratio:
\[ \frac{\text{Cooling Rate}_1}{\text{Cooling Rate}_2} = \frac{E_1 \cdot A_1 / (m_1 \cdot C_1)}{E_2 \cdot A_2 / (m_2 \cdot C_2)} \]
Substituting the ratios:
\[ = \frac{E_1}{E_2} \cdot \frac{A_1}{A_2} \cdot \frac{m_2}{m_1} \cdot \frac{C_2}{C_1} \]
Plugging in the values:
- \(E_1/E_2 = 3/1\)
- \(A_1/A_2 = 1/4\)
- \(m_2/m_1 = 8/1\)
- \(C_2/C_1 = 3/2\)

Therefore:
\[ \frac{\text{Cooling Rate}_1}{\text{Cooling Rate}_2} = \frac{3}{1} \cdot \frac{1}{4} \cdot \frac{8}{1} \cdot \frac{3}{2} \]

6. **Calculate the final ratio:**
\[ = \frac{3 \cdot 1 \cdot 8 \cdot 3}{1 \cdot 4 \cdot 1 \cdot 2} = \frac{72}{8} = 9 \]
This means:
\[ \text{Cooling Rate}_1 : \text{Cooling Rate}_2 = 9 : 1 \]

7. **Conclusion:**
Since the cooling rate of sphere 1 is greater than that of sphere 2, sphere 1 will cool at a faster rate initially.