A sample of gas in a balloon has an initial temperature of 25\({\rm \, ^{\circ}C}\) and a volume of 1670\({\rm L}\) . If the temperature changes to 82\({\rm \, ^{\circ}C}\) , and there is no change of pressure or amount of gas, what is the new volume, \(\texttip{V_{\rm 2}}{V_2}\), of the gas?

(V1/T1) = (V2/T2)

Remember T must be in kelvin.

To solve this problem, we can use the combined gas law equation:

\(\text{{P}}_1\text{{V}}_1/\text{{T}}_1 = \text{{P}}_2\text{{V}}_2/\text{{T}}_2\)

where:
P1 = initial pressure
V1 = initial volume
T1 = initial temperature
P2 = final pressure (assuming no change, so it can be taken as P1)
V2 = final volume (what we're trying to find)
T2 = final temperature

In this case, we are given:
T1 = 25°C = 298 K (convert temperature to Kelvin)
V1 = 1670 L
T2 = 82°C = 355 K (convert temperature to Kelvin)

Let's substitute these values into the combined gas law equation:

\(\text{{P}}_1 \times \text{{V}}_1/\text{{T}}_1 = \text{{P}}_2 \times \text{{V}}_2/\text{{T}}_2\)

Since the pressure is constant (no change), we can simplify the equation to:

\(\text{{V}}_1/\text{{T}}_1 = \text{{V}}_2/\text{{T}}_2\)

Now we can rearrange the equation to solve for V2:

\(\text{{V}}_2 = (\text{{V}}_1 \times \text{{T}}_2)/\text{{T}}_1\)

Plugging in the values:

\(\text{{V}}_2 = (1670 \, \text{{L}} \times 355 \, \text{{K}})/298 \, \text{{K}}\)

Calculating this expression will give us the final volume, V2.