To solve this problem, let's go through the steps carefully.

Given Data

The volume $V$ of a gas is directly proportional to its temperature $T$ and inversely proportional to the pressure $P$. Mathematically, this can be expressed as: $V = k \cdot \frac{T}{P}$ where $k$ is a constant of proportionality.

We are given:

1. $V = 70 \, \text{cm}^3$ when $T = 420 \, \text{K}$ and $P = 18 \, \text{kg/cm}^2$.
2. We need to find $T$ when $V = 60 \, \text{cm}^3$ and $P = 7 \, \text{kg/cm}^2$.

Step 1: Find the Constant $k$

Using the first set of values: $70 = k \cdot \frac{420}{18}$ Solving for $k$: $k = 70 \cdot \frac{18}{420} = \frac{70 \cdot 18}{420} = \frac{1260}{420} = 3$ So, $k = 3$.

Step 2: Use $k$ to Find $T$ with New Values of $V$ and $P$

Now that we know $k = 3$, we can substitute it along with $V = 60 \, \text{cm}^3$ and $P = 7 \, \text{kg/cm}^2$ into the formula to find $T$: $60 = 3 \cdot \frac{T}{7}$ Solving for $T$: $\frac{T}{7} = \frac{60}{3} = 20$ $T = 20 \times 7 = 140$

Answer

The temperature $T$ is $140$ Kelvin.

Would you like a deeper breakdown or have any questions on this?

Here are five related questions you might consider:

1. How would the formula change if the volume was inversely proportional to both temperature and pressure?
2. What happens to the temperature if both volume and pressure are doubled?
3. How does changing the proportionality constant $k$ affect the relationship?
4. Can you use this method to find the pressure given volume and temperature?
5. What assumptions are we making about the gas for these proportional relationships to hold?

Tip: In proportional relationships, understanding the constant of proportionality is key to finding how variables relate to each other across different conditions.