Question

You have \(3.5 \mathrm{L}\) of \(\mathrm{NO}\) at a temperature of \(22.0^{\circ} \mathrm{C} .\) What volume would the NO occupy at \(37^{\circ} \mathrm{C} ?\) (Assume the pressure is constant.)

Short answer

The final volume of NO at 37°C is approximately 3.674 L.

Step-by-step solution

Understand the Gas Law

We will use Charles's Law, which states that the volume of a gas is directly proportional to its temperature in Kelvin, if the pressure remains constant. The formula is \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( T_1 \) are the initial volume and temperature, respectively, and \( V_2 \) and \( T_2 \) are the final volume and temperature.

Convert Temperatures to Kelvin

Convert both temperatures from Celsius to Kelvin. The conversion formula is \( K = ^\circ C + 273.15 \).Initial temperature, \( T_1 = 22.0 + 273.15 = 295.15 \ K \)Final temperature, \( T_2 = 37.0 + 273.15 = 310.15 \ K \)

Apply Charles's Law

Substitute the known values into Charles's Law formula to find the final volume \( V_2 \).\[ \frac{3.5}{295.15} = \frac{V_2}{310.15} \]

Solve the Equation for V2

Solve the equation for \( V_2 \) by cross-multiplying and isolating \( V_2 \).\[ V_2 = \frac{3.5 \times 310.15}{295.15} \approx 3.674 \text{ L} \]

Key concepts

Introduction to Gas Laws

Gas laws are fundamental principles that describe how the volume, pressure, and temperature of a gas interact with one another. These laws are especially important for understanding the behavior of gases under various conditions. There are several key gas laws, including Boyle's law, Charles's law, and Avogadro's law, each explaining a different aspect of gas behavior. In this context, we focus on Charles's Law, which relates the volume and temperature of a gas when pressure is constant.
Understanding these laws helps chemists and physicists predict how gases will act in different environments, which is essential for developments in technology, industry, and even weather prediction.

• Boyle's Law: relates pressure and volume (at constant temperature)
• Charles's Law: relates temperature and volume (at constant pressure)
• Avogadro's Law: relates volume and the amount of gas (at constant temperature and pressure)

Volume-Temperature Relationship

The volume-temperature relationship of gases is expressed through Charles's Law. This principle asserts that the volume of a given mass of gas is directly proportional to its temperature when pressure remains constant. As a gas's temperature increases, its volume expands; conversely, the volume decreases as the temperature drops. This happens because an increase in temperature provides energy for gas molecules to move more vigorously, thus occupying more space.
The mathematical representation of this relationship is:
\[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]Here, \(V_1\) and \(T_1\) are the initial volume and temperature, and \(V_2\) and \(T_2\) are the final volume and temperature. It's important to measure temperature in Kelvin for accurate calculations, as this scale starts from absolute zero, at which molecular motion ceases.
This relationship highlights the fundamental kinetic theory of gases, emphasizing the dynamic nature of gas particles.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is essential in applying Charles's Law effectively. In this scale, temperatures are absolute, meaning they start at the theoretical point where there is no thermal energy, known as absolute zero. All equations involving gas laws use Kelvin because it provides consistency and precision.
To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature:

• Initially, if a gas is at \(22.0^\circ \text{C}\), convert it to Kelvin: \( T_1 = 22.0 + 273.15 = 295.15 \, \text{K} \)
• For a change in temperature to \(37.0^\circ \text{C}\), convert it as well: \( T_2 = 37.0 + 273.15 = 310.15 \, \text{K} \)

Calculating with Kelvin ensures that the proportionalities in Charles's Law are accurate, enabling precise predictions of volume changes with temperature fluctuations.
Remember, when converting temperatures for calculations, using Kelvin avoids negative values, maintaining the integrity of your results.