Question

If a 375 -mL sample of neon gas is heated from 24 'C to 72 ' \(\mathrm{C}\) at constant pressure, what will be the volume of the sample at the higher temperature?

Short answer

The final volume of the neon gas sample at 72 °C and constant pressure is approximately 436.5 mL. We calculate this by first converting the given temperatures from Celsius to Kelvin and then applying Charles's Law, (V1/T1)=(V2/T2), where V1 and T1 are the initial volume and temperature, respectively, and V2 and T2 are the final volume and temperature, respectively.

Step-by-step solution

Convert temperatures from Celsius to Kelvin

To convert temperatures from Celsius to Kelvin, we add 273.15 to the Celsius temperature. Initial temperature in Kelvin (T1): T1 = 24 °C + 273.15 = 297.15 K Final temperature in Kelvin (T2): T2 = 72 °C + 273.15 = 345.15 K

Apply Charles's Law and solve for final volume

Charles's Law states that the volume of a gas is directly proportional to its temperature, provided the pressure remains constant. Mathematically, it can be expressed as: (V1/T1)=(V2/T2) Where V1 and T1 are the initial volume and temperature, respectively, and V2 and T2 are the final volume and temperature, respectively. We are given the initial volume, V1 (375 mL), and we have calculated T1 and T2 using Step 1. Now we can solve for the final volume, V2: V2 = V1 * (T2 / T1) V2 = 375 mL * (345.15 K / 297.15 K) V2 ≈ 436.5 mL The final volume of the neon gas sample at 72 °C and constant pressure is approximately 436.5 mL.

Key concepts

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases in response to changes in pressure, volume, and temperature. One key aspect of these laws is their ability to predict the behavior of a gas under varying conditions, which is crucial in fields such as chemistry, physics, and engineering.

The cornerstone of gas law is the idea that gases tend to expand when heated and compress when cooled, all the while exerting pressure on their container. This expansion and compression can be quantified by specific gas laws, such as Boyle's Law (pressure-volume relationship), Charles's Law (temperature-volume relationship), and Gay-Lussac's Law (pressure-temperature relationship). When combined, these individual laws form the Combined Gas Law, which provides a comprehensive equation to account for simultaneous changes in pressure, volume, and temperature. Furthermore, when the number of molecules (or moles) of a gas is also factored in, the Ideal Gas Law emerges, encapsulating all of the separate gas laws into one overarching principle.

Temperature-Volume Relationship

The temperature-volume relationship of a gas is elegantly demonstrated by Charles's Law. This law states that the volume of a given amount of gas held at constant pressure is directly proportional to its Kelvin temperature. In other words, as the temperature of a gas increases, so does its volume, and vice versa.

When we talk about temperature in the context of gas laws, it's vital to use the Kelvin scale since it starts at absolute zero, the point at which particles theoretically have no kinetic energy and thus no motion. Utilizing temperatures in Kelvin allows us to apply Charles's Law correctly and avoid negative volumes, which don't make physical sense in this context. To better understand the concept in practice, a real-world example is helpful: Imagine a balloon exposed to heat, as the air inside the balloon warms up, its volume increases, causing the balloon to expand. This principle is utilized in technologies like hot air balloons and is an everyday demonstration of the temperature-volume relationship.

Kelvin Temperature Scale

The Kelvin temperature scale is an absolute temperature scale, meaning it begins at absolute zero, where all molecular motion stops. The scale is named after Lord Kelvin (William Thomson), who developed it in the mid-1800s. It's particularly important in the study of thermodynamics and gas laws because the Kelvin scale provides a direct measurement of the thermal energy within a substance.

The key distinction between the Kelvin scale and the Celsius or Fahrenheit scales is that the Kelvin scale does not have negative numbers. This characteristic simplifies the mathematical relationships in the gas laws, such as Charles's Law, making it the preferable scale for scientific calculations. To convert from Celsius to Kelvin, one simply adds 273.15 to the Celsius temperature, thus shifting the starting point from the freezing point of water to absolute zero. For example, the boiling point of water, 100 °C, is equivalent to 373.15 K in the Kelvin scale, and this consistent conversion is essential in ensuring accuracy when dealing with gas behavior at different temperatures.