The temperature of a gas filled in closed vessel reduces by `1^@ C`. Calculate the ratio of change in pressure to the original pressure of the gas.

Explanation

To solve the problem of calculating the ratio of change in pressure to the original pressure of a gas when its temperature decreases by \(1^\circ C\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between pressure and temperature:**
According to the ideal gas law, pressure (P) is directly proportional to temperature (T) when the volume and the number of moles of gas are kept constant. This can be expressed mathematically as:
\[ P \propto T \]

2. **Express the relationship in terms of changes:**
We can express the change in pressure (\(dP\)) in relation to the original pressure (\(P\)) and the change in temperature (\(dT\)) in relation to the original temperature (\(T\)):
\[ \frac{dP}{P} = k \cdot \frac{dT}{T} \]
where \(k\) is a constant that depends on the relationship between pressure and temperature.

3. **Identify the change in temperature:**
The problem states that the temperature decreases by \(1^\circ C\). In Kelvin, this change is also \(1 K\) since a change in temperature is the same in both Celsius and Kelvin scales. Therefore:
\[ dT = -1 \text{ (since it is a decrease)} \]

4. **Assume original temperature:**
Let the original temperature be \(T\).

5. **Substituting values into the equation:**
Now, substituting \(dT = -1\) into our equation gives:
\[ \frac{dP}{P} = k \cdot \frac{-1}{T} \]
Since we know that pressure is directly proportional to temperature, we can take \(k = 1\) for simplicity:
\[ \frac{dP}{P} = -\frac{1}{T} \]

6. **Final expression for the ratio:**
Therefore, the ratio of the change in pressure to the original pressure is:
\[ \frac{dP}{P} = -\frac{1}{T} \]

### Conclusion:
The ratio of change in pressure to the original pressure of the gas when the temperature decreases by \(1^\circ C\) is:
\[ \frac{dP}{P} = -\frac{1}{T} \]