This problem can be solved using the Ideal Gas Law, which is represented by the equation:

$PV = nRT$

Where:

• $P$ is the pressure of the gas
• $V$ is the volume of the gas
• $n$ is the number of moles of gas
• $R$ is the universal gas constant
• $T$ is the absolute temperature in Kelvin

We are given two different sets of conditions (before and after the journey), and we need to find the volume $V_2$ after the journey. Since no air was lost during the journey, the number of moles $n$ remains constant. We can therefore use the combined gas law:

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

Where:

• $P_1 = 250 \, \text{kPa}$ (initial pressure)
• $V_1 = 12.0 \, \text{dm}^3$ (initial volume)
• T_1 = 10 \, ^\circ C = 10 + 273.15 = 283.15 \, \text{K} (initial temperature)
• $P_2 = 261 \, \text{kPa}$ (final pressure)
• T_2 = 25 \, ^\circ C = 25 + 273.15 = 298.15 \, \text{K} (final temperature)
• $V_2$ is the unknown final volume.

Let's rearrange the combined gas law to solve for $V_2$:

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

Now, let's plug in the values and calculate the final volume.The final volume of the tyre after the journey is approximately $12.10 \, \text{dm}^3$.

Would you like a detailed explanation of the calculation, or do you have any questions?

Here are 5 related questions you might find helpful:

1. How does the Ideal Gas Law relate pressure, volume, and temperature?
2. What assumptions are made when using the Ideal Gas Law?
3. How would the calculation change if there was air loss during the journey?
4. What is the significance of converting temperature to Kelvin in gas law calculations?
5. How does an increase in temperature affect the pressure and volume of a gas?

Tip: Always convert temperatures to Kelvin when using the Ideal Gas Law to avoid errors in calculations!