Problem 24 - Master Resource Book in JEE Main Physics

Question

A cubic vessel (with faces horizontal + vertical) contains an ideal gas at NTP. The vessel is being carried by a rocket which is moving at a speed of \(500 \mathrm{~ms}^{-1}\) in vertical direction. The pressure of the gas inside the vessel as observed by us on the ground [NCERT Exemplar]

Step-by-Step Solution

Answer

The observed pressure is the same, 101.3 kPa.

Step 1: Understanding Normal Temperature and Pressure (NTP)

At NTP, the pressure of a gas is defined as 1 atm or 101.3 kPa and the temperature is 0°C or 273.15 K. These conditions apply to the gas inside the cubic vessel initially.

Step 2: Considering Relative Motion Effects

The rocket's motion might suggest there could be a relativistic effect on pressure. However, the speed of 500 m/s is much less than the speed of light, so relativistic effects are negligible.

Step 3: Analyzing the Pressure in the Moving Rocket

According to the principles of physics, specifically the independence of pressure measurement from uniform motion (Galilean invariance for classical mechanics), the pressure inside the vessel remains unchanged as observed from the ground.

Step 4: Conclusion on Observed Pressure from the Ground

Since the speed of the rocket and the absence of relativistic effects suggests no change in pressure, the pressure observed from the ground remains the same as the starting condition at NTP, which is 101.3 kPa.

Key Concepts

NTP (Normal Temperature and Pressure)

Normal Temperature and Pressure, commonly abbreviated as NTP, is a standard condition for measuring the properties of gases. It provides a baseline temperature and pressure to facilitate comparisons. At NTP, the temperature is set at 0°C, which converts to 273.15 Kelvin. The pressure is defined as 1 atmosphere, equivalent to 101.3 kPa. By using these standard measures, scientists and engineers can accurately compare different gases under identical conditions.

• Standard Temperature: 273.15 K (0°C)
• Standard Pressure: 101.3 kPa (1 atm)

This concept is especially crucial when considering gases in varied environments, such as within our cubic vessel in the original exercise.

Pressure Measurement

Pressure measurement involves determining the force applied by a gas on the walls of its container. For the cubic vessel containing an ideal gas at NTP, pressure remains constant under uniform motion, as in the moving rocket scenario.

• Pressure is defined as force per unit area.
• At NTP, this pressure is 101.3 kPa, regardless of external motion, assuming non-relativistic speeds.

When measuring pressure, it's also important to account for any external variables which might affect the gas, such as temperature changes or additional forces, but in this case, the speed of the rocket (500 m/s) does not contribute to any noticeable changes in pressure.

Galilean Invariance

Galilean Invariance is a principle in classical physics stating that the laws of motion are the same in any inertial frame. That is to say, if a system moves with a constant velocity, the fundamental behaviors—such as pressure in a gas—remain unchanged relative to an outside observer.

• Independent of uniform motion, pressure in a gas is not affected by the motion of its container.
• Thus, for the cubic vessel on a rocket, the observed pressure from the ground remains at 101.3 kPa.

In practical terms, this means that although the rocket moves vertically at 500 m/s, such speed is too low to introduce relativistic effects that might alter the pressure inside the vessel. Hence, the principle confirms that the gas's pressure inside the rocket will be seen as unchanged from the perspective of someone on the ground.