A sample of helium at a pressure of 745 torr and in a volume of \(2.58 \mathrm{~L}\) was heated from 24.0 to \(75.0^{\circ} \mathrm{C}\). The volume of the container expanded to \(2.81 \mathrm{~L}\). What was the final pressure (in torr) of the helium?

Understanding gas law calculations is crucial to solving problems related to the behavior of gases under various conditions of pressure, volume, and temperature. According to the combined gas law, the relationship between these variables can be represented with the equation \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\), where \(P\) stands for pressure, \(V\) for volume, and \(T\) for temperature. The subscripts 1 and 2 denote the initial and final states of the gas, respectively. To solve these equations, it's important to use consistent units; typically, pressure is in atmospheres (atm), volume in liters (L), and temperature in Kelvin (K).

To find the final pressure \(P_2\) in the provided example, one would rearrange the combined gas law to isolate \(P_2\) and then plug in the known values, making sure to convert all the measurements to appropriate units before doing the calculations. This type of calculation enables us to predict how a gas will behave when it undergoes changes in conditions, which is fundamental in fields like chemistry and physics.

Temperature and pressure are key factors in the study of gases, and converting them correctly is important for gas law calculations. Pressure, the force exerted by a gas per unit area, can be measured in various units, with atmospheres (atm) and torr being common ones used in the laboratory. Conversion between these two units is straightforward: 1 atm is equal to 760 torr. This relationship is especially useful when dealing with gas laws, as many equations require pressure to be in atmospheres.

When solving problems related to gas behavior, it's essential to first convert the given pressure to the units required by the equation in use. For example, in the exercise, the initial pressure measured in torr was converted to atmospheres before applying the combined gas law. This meticulous approach to unit conversion is necessary to prevent errors in the final calculation.

The combined gas law also involves temperature, which must be in Kelvin for the law to be applicable. The Kelvin scale is an absolute temperature scale, meaning it starts from absolute zero, the lowest possible temperature. This scale is key in scientific calculations because it avoids negative numbers for temperature, which can complicate gas law calculations.

To convert Celsius to Kelvin, which is common in gas law problems, the formula used is \( T(K) = T(\circ C) + 273.15 \). This conversion is paramount because Celsius and Kelvin scales have different starting points (0 °C is not the lowest possible temperature, while 0 K is). In our provided exercise, both the initial and final temperatures given in Celsius are converted to Kelvin before inserting them into the combined gas law formula.

The behavior of gases under different conditions of temperature, pressure, and volume is predictable using the gas laws. These laws were derived from empirical observations and can be combined into the combined gas law, which is a versatile equation that incorporates Boyle's Law, Charles's Law, and Gay-Lussac's Law. The combined gas law shows that if the amount of gas and the amount of moles remain constant, the pressure, volume, and temperature of a gas are interrelated.

For gases, pressure increases as temperature increases, provided the volume is constant. If the volume can expand, the rise in temperature may not cause an increase in pressure. Moreover, increasing the volume of a container while keeping temperature constant leads to reduced pressure. These fundamental principles about the behavior of gases are essential for understanding how to manipulate and predict the state of a gas in different scenarios, such as in the example exercise where the volume expansion and temperature increase led to calculating the new pressure of helium.