E Consider an ideal gas that exerts a pressure of \(23.76 \mathrm{~mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C}\). Assuming \(n\) and \(V\) are held constant, what would its pressure be at \(40^{\circ} \mathrm{C}\) ? \(70^{\circ} \mathrm{C}^{2} 100^{\circ} \mathrm{C}\) ? Compare the numbers you have just calculated with the vapor pressures of water at these temperatures. Can you suggest a reason why the two sets of numbers are so different?

Charles's Law holds a fundamental place in the study of gases and thermodynamics. It is an empirical law that describes how gases tend to expand when heated. In a concise form, Charles's Law states that, for an ideal gas at constant pressure, the volume of the gas is directly proportional to its absolute temperature (in Kelvin).

The relationship is commonly expressed as: \( V \propto T \) or \( \frac{V1}{T1} = \frac{V2}{T2} \), where \( V \) refers to the volume, and \( T \) denotes the temperature in Kelvin. This equation tells us that if you increase the temperature of a gas (while keeping the pressure constant), the volume will increase, and vice versa. When solving problems with Charles's Law, it's crucial to convert all temperature readings to Kelvin, as this is the standard unit for thermodynamic temperature scale in physics.

Intuition Behind Charles's Law

Imagine you have a balloon filled with air. As you heat the air inside the balloon, the particles move faster and push outward more forcefully, causing the balloon to expand. Conversely, if you cool it down, the particle movement slows, and the balloon shrinks. It's these principles that students can visualize to understand the behavior of gases under temperature changes.

Almost all thermodynamic calculations require temperature to be in the Kelvin scale. The Kelvin scale is an absolute thermodynamic temperature scale, and it is the basis for the International System of Units (SI). The conversion from Celsius to Kelvin is quite straightforward: \( T_{Kelvin} = T_{Celsius} + 273.15 \).

For instance, in the provided exercise, the temperature must be converted into Kelvin before applying Charles's Law. This is because the Kelvin scale starts at absolute zero, the point at which there is theoretically no particle movement, and provides a direct proportionality between volume and temperature which is the cornerstone of Charles's Law.

Students often overlook the importance of using the correct temperature units, resulting in errors. Remember that Celsius and Fahrenheit are not suitable for gas law calculations unless the conversion to Kelvin is performed first.

Pressure is a critical concept in gas laws, defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. In the context of gases, it's the force that gas molecules exert when they collide with the walls of their container.

In our exercise, we convert pressure from millimeters of mercury (mmHg) to Pascals (Pa) since Pascal is the SI unit for pressure. The conversion factor is important: \( 1 \text{ mmHg} = 133.322 \text{ Pa} \).

Understanding the Conversion

Imagine the pressure as the weight of the column of air above a particular surface. The weight creates a force pressing down on that surface. When we talk about millimeters of mercury, we're describing this pressure by how high it would raise a column of mercury. Converting this measurement to Pascals allows us to use it within the international standard unit system, ensuring conformity and comparability of data.

Understanding vapor pressure—the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system—is key in thermodynamics and physical chemistry.

Comparing the calculated pressures of the ideal gas with the vapor pressures of water, we see a pronounced difference, often attributed to several factors. Vapor pressure of a substance depends on its unique physical properties. For instance, water has high vapor pressure at relatively low temperatures because of its ability to evaporate easily.

Substances vary in how readily they vaporize, known as volatility. A substance with higher volatility has a higher vapor pressure at the same temperature compared to a less volatile substance. This concept helps explain why we see large disparities between the pressures in this exercise; water is more volatile and has stronger intermolecular attractions than many other substances, affecting its vapor pressure.