5. The pressure of a gas is inversely proportional to its volume and directly proportional to its absolute temperature. When the pressure is $$ 10^{6} $$ pa and the temperature $$ 300^{\circ} \mathrm{K} $$, the volume is $$ 1000 \mathrm{~cm}^{3} $$. Find the volume when the pressure is $$ 1.8 imes 10^{6} $$ pa and the temperature $$ 324^{\circ} \mathrm{K} $$.

Question

5. The pressure of a gas is inversely proportional to its volume and directly proportional to its absolute temperature. When the pressure is $$ 10^{6} $$ pa and the temperature $$ 300^{\circ} \mathrm{K} $$, the volume is $$ 1000 \mathrm{~cm}^{3} $$. Find the volume when the pressure is $$ 1.8 	imes 10^{6} $$ pa and the temperature $$ 324^{\circ} \mathrm{K} $$.

UpStudy AI · Accepted Answer

To solve the problem, we'll use the relationship between pressure $$ P $$, volume $$ V $$, and temperature $$ T $$ given by:

$$
P \propto \frac{T}{V}
$$

This can be expressed as:

$$
P = k \frac{T}{V}
$$

where $$ k $$ is the proportionality constant.

### Step 1: Establish the Relationship

Given two states of the gas:
1. **Initial State:**
   - Pressure $$ P_1 = 10^6 $$ Pa
   - Temperature $$ T_1 = 300 $$ K
   - Volume $$ V_1 = 1000 $$ cm³

2. **Final State:**
   - Pressure $$ P_2 = 1.8 \times 10^6 $$ Pa
   - Temperature $$ T_2 = 324 $$ K
   - Volume $$ V_2 = ? $$

Using the proportionality:

$$
\frac{P_1}{P_2} = \frac{T_1 / V_1}{T_2 / V_2}
$$

Rearranging to solve for $$ V_2 $$:

$$
V_2 = V_1 \times \frac{P_1 \times T_2}{P_2 \times T_1}
$$

### Step 2: Plug in the Known Values

$$
V_2 = 1000 \, \text{cm}^3 \times \frac{10^6 \, \text{Pa} \times 324 \, \text{K}}{1.8 \times 10^6 \, \text{Pa} \times 300 \, \text{K}}
$$

### Step 3: Simplify the Calculation

$$
V_2 = 1000 \times \frac{324}{1.8 \times 300} = 1000 \times \frac{324}{540} = 1000 \times 0.6 = 600 \, \text{cm}^3
$$

### **Final Answer:**

When the pressure is $$ 1.8 \times 10^6 $$ Pa and the temperature is $$ 324 $$ K, the volume of the gas is **600 cm³**.
When the pressure is $$ 1.8 \times 10^6 $$ Pa and the temperature is $$ 324 $$ K, the volume is **600 cm³**.

The pressure of a gas is inversely
proportional to its volume and directly
proportional to its absolute temperature.
When the pressure is pa and the
temperature , the volume is
. Find the volume when the
pressure is pa and the
temperature .

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The pressure of a gas is inversely proportional to its volume and directly proportional to its absolute temperature. When the pressure is pa and the temperature , the volume is . Find the volume when the pressure is pa and the temperature .