22 g of carbon dioxide at `27^(@)C` is mixed in a closed container with 16 g of oxygen at `37^(@)C`. If both gases are considered as ideal gases, then the temperature of the mixture is

Options

A. `24.2^(@)C`
B. `28.5^(@)C`
C. `32^(@)C`
D. `33.5^(@)C`

Answer

To solve the problem of finding the final temperature of a mixture of carbon dioxide and oxygen gases, we can follow these steps:

### Step 1: Calculate the number of moles of each gas
We start by calculating the number of moles of carbon dioxide (CO₂) and oxygen (O₂).

- For CO₂:
- Mass of CO₂ = 22 g
- Molar mass of CO₂ = 44 g/mol
- Number of moles (N₁) of CO₂ = Mass / Molar mass = 22 g / 44 g/mol = 0.5 mol

- For O₂:
- Mass of O₂ = 16 g
- Molar mass of O₂ = 32 g/mol
- Number of moles (N₂) of O₂ = Mass / Molar mass = 16 g / 32 g/mol = 0.5 mol

### Step 2: Convert temperatures to Kelvin
Next, we convert the given temperatures from Celsius to Kelvin.

- Temperature of CO₂ (T₁) = 27°C = 27 + 273 = 300 K
- Temperature of O₂ (T₂) = 37°C = 37 + 273 = 310 K

### Step 3: Apply the principle of conservation of energy
According to the principle of conservation of energy, the total internal energy before mixing is equal to the total internal energy after mixing.

The equation can be expressed as:
\[ N₁ \cdot C_v \cdot T₁ + N₂ \cdot C_v \cdot T₂ = (N₁ + N₂) \cdot C_v \cdot T_f \]
Where:
- \(T_f\) is the final temperature of the mixture.
- \(C_v\) is the molar specific heat at constant volume (which cancels out).

### Step 4: Rearranging the equation
We can simplify the equation by canceling \(C_v\):
\[ N₁ \cdot T₁ + N₂ \cdot T₂ = (N₁ + N₂) \cdot T_f \]
Rearranging gives:
\[ T_f = \frac{N₁ \cdot T₁ + N₂ \cdot T₂}{N₁ + N₂} \]

### Step 5: Substitute the values
Substituting the values we calculated:
\[ T_f = \frac{(0.5 \cdot 300) + (0.5 \cdot 310)}{0.5 + 0.5} \]
\[ T_f = \frac{150 + 155}{1} = 305 \text{ K} \]

### Step 6: Convert final temperature back to Celsius
Finally, we convert the final temperature from Kelvin back to Celsius:
\[ T_f = 305 \text{ K} - 273 = 32 \text{ °C} \]

### Final Answer
The final temperature of the mixture is **32 °C**.

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