A \(0.20-\mathrm{g}\) sample of a polymer, dissolved in \(0.100 \mathrm{~L}\) of toluene, has an osmotic pressure of \(6.3\) Torr at \(20 .{ }^{\circ} \mathrm{C}\). What is the molar mass of the polymer?

Understanding the concept of osmotic pressure is key when dealing with solutions and it's particularly important in determining the properties of polymers in solution. Osmotic pressure, represented as \( \Pi \), is a measure of the tendency of a solvent to move into a solution through a semi-permeable membrane.
In a practical sense, when a polymer is dissolved in a solvent, like in our exercise, it creates a solution that has a different osmotic pressure than the pure solvent. This pressure can be calculated using the equation \( \Pi = MRT \) where \( M \) stands for molarity (moles of solute per liter of solution), \( R \) is the ideal gas constant (0.0821 L atm / (mol K)), and \( T \) is the absolute temperature in Kelvin.
Now, let’s observe the process of solving for osmotic pressure. Firstly, convert the given pressure from Torr to atm because standard equations use the latter. To do so, divide the pressure value by 760, since 760 Torr equals 1 atm. Next, plug the converted pressure, ideal gas constant, and temperature (converted to Kelvin) into the osmotic pressure equation and solve for molarity.
This calculation is instrumental for scientists and engineers as it helps in determining very small molar masses of polymers and proteins - something that other techniques might not do as precisely.

Temperature conversion is a basic necessity in various scientific calculations, including those of chemistry and physics. It's important to note that temperatures in equations such as the one for osmotic pressure must be in Kelvin. The Kelvin scale is an absolute temperature scale, starting at absolute zero, the point at which all molecular motion ceases.
To convert Celsius to Kelvin, which is required for our polymer exercise, you simply add 273.15 to the Celsius temperature. The formula is: \( T(K) = T(^\circ C) + 273.15 \). For instance, our exercise provided a temperature of 20°C, which we convert to Kelvin by adding 273.15, resulting in 293.15 K.
Always remember when solving chemical problems to ensure that your temperature is in the right unit, as forgetting to convert to Kelvin may lead to incorrect calculations.

The concept of molarity is pivotal in chemistry; it's the concentration of a solution expressed as the number of moles of solute per liter of solution. To determine molarity, you can use the osmotic pressure as done in our polymer example.
With the formula \( M = \frac{\Pi}{RT} \), you can find the molarity once the osmotic pressure (in atm) and temperature (in Kelvin) are known. After finding the molarity, it can be used to calculate the number of moles by multiplying it by the volume of the solution (in liters), as molarity is moles per liter.
Once you have the moles, the molar mass of the polymer can be determined by dividing the mass of the polymer sample (in grams) by the number of moles. This step reveals the molar mass, which is a vital characteristic of polymers, affecting their physical and chemical properties significantly. Understanding how to calculate molarity is not only important for this reason but also because it’s a foundational concept in preparing solutions and performing titrations in the lab.