To solve this problem, we can use the combined gas law, which relates the initial and final conditions of pressure, volume, and temperature for a given amount of gas.
The combined gas law is given by:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
Where: P1 = initial pressure V1 = initial volume T1 = initial temperature P2 = final pressure V2 = final volume T2 = final temperature
Let's assign the given values to the variables:
P1 = 760 mmHg V1 = 2.5 L T1 = 473 K P2 = 1140 torr V2 = 1.75 L T2 = unknown (to be calculated)
We need to make sure that the pressures and volumes are expressed in the same units. Let's convert the pressure from torr to mmHg:
1 torr = 1 mmHg
Now we can substitute the values into the combined gas law equation and solve for T2:
(760 mmHg * 2.5 L) / 473 K = (1140 mmHg * 1.75 L) / T2
To solve for T2, we can cross-multiply and then divide both sides:
(760 mmHg * 2.5 L * T2) = (1140 mmHg * 1.75 L * 473 K)
T2 = (1140 mmHg * 1.75 L * 473 K) / (760 mmHg * 2.5 L)
T2 ≈ 722.08 K
Therefore, the temperature needed to reduce the volume to 1.75 L at 1140 torr is approximately 722.08 K.