What is the molar mass of a gas if \(0.0494 \mathrm{g}\) of the gas occupies a volume of \(0.100 \mathrm{L}\) at a temperature \(26^{\circ} \mathrm{C}\) and a pressure of 307 torr?

The Ideal Gas Law is a cornerstone of chemical thermodynamics, offering a quick way to relate the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. This equation is as follows: \(PV = nRT\). Here, R represents the ideal gas constant, which bridges the units of pressure, volume, temperature, and amount of substance. This law permits the solving of various problems, such as finding the molar mass of a gas.

It is vital to plug in the right units into the equation. Commonly, the units for pressure, volume, and temperature are atmospheres (atm), liters (L), and kelvins (K), respectively. The value of R should correspond with these units. In exercises that require finding the molar mass, the Ideal Gas Law enables the calculation of the number of moles, which is the quantity directly related to the molar mass through the simple formula: \(\text{Molar mass} = \frac{\text{mass}}{\text{number of moles}}\).

When using the Ideal Gas Law, temperature must be in Kelvin, the SI base unit for thermodynamic temperature. This requirement stems from Kelvin's absolute scale, which is directly proportional to the energy per particle due to thermal motion. To convert from Celsius to Kelvin, one simply adds 273.15 to the Celsius value. The conversion is significant since using degrees Celsius would not afford you the proportional relationship between temperature and kinetic energy in the Ideal Gas Law.

For example, if our given temperature is 26°C, converting it to Kelvin would be as follows: \(T(K) = 26°C + 273.15 = 299.15 K\). Always remember this step, as omitting it can lead to incorrect results and misunderstandings of the behavior of gases.

The gas constant, R, is a bridge between various physical properties and an essential part of the Ideal Gas Law equation. It unifies the pressure, volume, temperature, and mole count allowing for the solution of chemical problems involving gases. The value of R depends on the units used for pressure, volume, and temperature.

For calculations involving atm, L, and K, a common value for R is \(0.0821 L\cdot atm/(mol\cdot K)\). It's important to ensure consistency in units throughout a problem to avoid confusion and errors. If pressure is given in torr or mmHg, for example, it is important to convert these units to atmospheres or adjust the gas constant accordingly to align with pressure's units.

Pressure conversion to atmospheres is necessary when solving problems using the Ideal Gas Law because the constant R's most common value is defined in terms of atmospheres. To convert torr to atmospheres, divide the pressure in torr by 760, since there are 760 torr in one atmosphere.

For example, if the pressure of a gas is given as 307 torr, convert it to atmospheres like this: \(P(atm) = 307 torr \times (1 atm / 760 torr) = 0.403947368 atm\). This consistent use of units is foundational to achieving correct results. Precision in such conversions is one reason why attention to detail in following steps of calculations in chemistry is necessary.