Question

Scientists calculate the pressure within a gas by using the following equation: $$ P=\frac{N \times k \times T}{V} $$ In the equation: \(P\) is the pressure of the gas; \(N\) is the number of particles in the gas; \(k\) is a constant; \(T\) is the temperature of the gas; \(V\) is the volume of the gas. If the number of particles in the gas decreases, which of the following changes will result in an increase in the pressure of the gas? A. decreasing both the volume and the temperature of the gas B. increasing both the volume and the temperature of the gas C. increasing the volume and decreasing the temperature of the gas D. decreasing the volume and increasing the temperature of the gas

Short answer

Option D: decreasing the volume and increasing the temperature of the gas would increase the pressure of the gas while also considering the decrease in the number of particles.

Step-by-step solution

Option A

Decreasing both the volume and the temperature of the gas. Since the number of particles decreases, in order to maintain the same pressure, the ratio of the temperature and volume should remain constant. However, in this option, we decrease both the temperature and volume. This would cause the pressure to decrease as well as the ratio \(\frac{T}{V}\) would decrease overall, which is not what we are looking for.

Option B

Increasing both the volume and the temperature of the gas. In this case, we increase both the temperature and volume, so the ratio of temperature and volume is likely to change. Since both sides are increasing, the ratio \(\frac{T}{V}\) must increase to compensate for the decrease in the number of particles. Since the pressure formula is directly proportional to the temperature and inversely proportional to the volume, this option could potentially result in an increase in the pressure.

Option C

Increasing the volume and decreasing the temperature of the gas. In this case, the temperature decreases and the volume increases. The pressure of the gas will decrease as the ratio \(\frac{T}{V}\) decreases. This option will not result in an increase in the pressure of the gas.

Option D

Decreasing the volume and increasing the temperature of the gas. In this option, the volume decreases and the temperature increases. This affects the ratio \(\frac{T}{V}\) as it increases. Since the pressure of the gas is directly proportional to the temperature and inversely proportional to the volume, this option will result in an increase in the pressure of the gas. Considering the above analysis, the answer is

Option D

decreasing the volume and increasing the temperature of the gas would increase the pressure of the gas while also considering the decrease in the number of particles.

Key concepts

Pressure Equation

The calculation of gas pressure is fundamental in understanding how gases behave under different conditions. The pressure equation given by \(P = \frac{N \times k \times T}{V}\) is a powerful tool in thermodynamics. It tells us how pressure \(P\) is related to the number of particles \(N\), Boltzmann's constant \(k\), temperature \(T\), and volume \(V\).

- **Number of Particles \(N\):** Directly affects the pressure. More particles mean more collisions within the container, increasing the pressure.- **Constant \(k\):** Represents Boltzmann's constant, crucial for scaling the equation correctly based on particle energy.- **Temperature \(T\):** Increases the kinetic energy of particles, leading to more vigorous collisions and thus higher pressure.- **Volume \(V\):** An increase in volume gives particles more space, reducing the frequency of collisions and lowering the pressure.

Overall, understanding this equation helps determine how changing one parameter while holding others constant will affect the overall gas pressure.

Temperature and Volume Relation

The relationship between temperature and volume is an essential concept in gas behavior. When analyzing gas pressure changes, understanding how temperature and volume interact is crucial.

**Charles’s Law** states that for a given amount of gas at constant pressure, the volume of the gas is directly proportional to its temperature in Kelvin. This means, as the temperature increases, the volume also increases if the pressure is constant.

In scenarios where the number of particles changes, adjusting temperature and volume requires careful consideration:

• Increasing temperature tends to increase the speed of particles, leading to an attempt to occupy larger space if not constrained by volume.
• Decreasing volume confines the space available for the gas molecules, which can increase pressure if temperature rises.

Adjusting volume and temperature while keeping an eye on particle quantity can effectively manipulate gas pressure.

Particle Quantity Effect

The number of particles \(N\) within a gas plays a major role in determining pressure. Fewer particles generally mean fewer collisions within the container, resulting in decreased pressure.

To counteract a decrease in particle number while maintaining or increasing pressure, you can:

• **Decrease Volume:** Reducing the volume forces remaining particles to collide more frequently with container walls, raising pressure.
• **Increase Temperature:** Raising temperature increases the energy and speed of the particles, causing more intense collisions.

Both methods adjust the gas's pressure in the presence of fewer particles, demonstrating their significant influence over a gas's behavior. Understanding this concept helps predict how changes in particle count affect conditions within a gas system.

Thermodynamics Concepts

Thermodynamics involves the principles that govern the relationships between different properties of gases. In this context, the dynamics between pressure, volume, and temperature as they relate to thermodynamic laws are key.

**Boyle’s Law** provides insights when temperature is constant, stating that pressure and volume are inversely proportional. This means that increasing volume decreases pressure, assuming temperature remains constant.

Equally important is **Gay-Lussac’s Law**, which relates pressure directly to temperature when volume is constant, implying that increasing temperature will increase pressure if the volume does not change.

Understanding these thermodynamic concepts allows you to predict how varying one property, such as pressure, influences others and the overall behavior of gases under different experimental or natural conditions.