Which of the following indicates Charles' law mathematically (when n, P are constant)?
1) VT = constant
2) `V_(1) = V_(0) (1 + 1/273)`
3) `V_(0) = V_(1)(1+ 1/273)`
4) V/T = constant (when n,P are constant)

Options

A. 1,3
B. 1,2
C. 2,3
D. 2,4

Answer

To solve the question regarding which of the following indicates Charles' law mathematically when the number of moles (n) and pressure (P) are constant, we can analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Charles' Law**:
Charles' Law states that the volume of a gas is directly proportional to its absolute temperature (in Kelvin) when the pressure and the number of moles of the gas are held constant. Mathematically, this can be expressed as:
\[ V \propto T \quad \text{or} \quad \frac{V}{T} = k \]
where \(k\) is a constant.

2. **Analyzing the Options**:
- **Option 1: \(VT = \text{constant}\)**: This is not a correct representation of Charles' Law. It suggests a product of volume and temperature is constant, which is not the case in Charles' Law.

- **Option 2: \(V_1 = V_0(1 + \frac{1}{273})\)**: This expression indicates that for a temperature increase of 1°C, the volume increases by \(\frac{1}{273}\) of the volume at 0°C. This aligns with the concept of Charles' Law, making this option correct.

- **Option 3: \(V_0 = V_1(1 + \frac{1}{273})\)**: This expression is incorrect because it suggests a reverse relationship that does not correctly represent the law. It implies that the initial volume is dependent on the final volume in a way that contradicts the direct proportionality of volume to temperature.

- **Option 4: \(\frac{V}{T} = \text{constant}\)**: This is a direct mathematical representation of Charles' Law, confirming that volume divided by temperature remains constant when pressure and the number of moles are constant. This option is also correct.

3. **Conclusion**:
The correct options that indicate Charles' Law mathematically are:
- **Option 2**: \(V_1 = V_0(1 + \frac{1}{273})\)
- **Option 4**: \(\frac{V}{T} = \text{constant}\)

### Final Answer:
The correct answers are **2 and 4**.

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