Nitrogen Gas In Cylinder: Calculating Final State

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Hey guys! Ever wondered what happens when you heat nitrogen gas inside a cylinder? Let's dive into a cool problem involving thermodynamics and figure it out step by step. This is a classic example that combines concepts from physics and chemistry, so buckle up!

Setting up the Problem

Okay, so we have a fixed volume of nitrogen gas (N₂) trapped inside a cylinder. This cylinder has a piston, which means the gas can expand or contract if needed, but in this case, the volume remains constant. Initially, the gas is at a pressure of 398 kPa and a temperature of 26.8°C. Now, we introduce an electric heater inside the cylinder. This heater is going to pump energy into the system, increasing the gas's internal energy and, consequently, its temperature and pressure. The heater draws a current of 1.9 amps (A) for a duration of 4.8 minutes.

Initial Conditions

Let’s break down what we know initially:

  • Gas: Nitrogen (N₂)
  • Initial Pressure (P₁): 398 kPa
  • Initial Temperature (T₁): 26.8°C
  • Current (I): 1.9 A
  • Time (t): 4.8 minutes

The main goal here is to find the final state of the nitrogen gas after the heating process. This usually means determining the final pressure (P₂) and final temperature (T₂). To do this, we’ll need to use some fundamental principles of thermodynamics and make a few assumptions to simplify the problem. This involves understanding how the electrical energy supplied by the heater translates into thermal energy within the gas, and how this affects the gas's state variables. It's like figuring out how much the gas will 'react' to the extra energy we're giving it!

Key Concepts and Assumptions

Before we jump into the calculations, let's highlight some key concepts and assumptions we’ll use:

  1. Ideal Gas Law: We'll assume nitrogen gas behaves like an ideal gas. This means we can use the ideal gas law, which states: PV = nRT, where:

    • P = Pressure
    • V = Volume
    • n = Number of moles
    • R = Ideal gas constant
    • T = Temperature

    The ideal gas law is a cornerstone in thermodynamics, especially when dealing with gases at moderate pressures and temperatures. It allows us to relate pressure, volume, temperature, and the amount of gas in a straightforward manner. This assumption simplifies the calculations significantly, as real gases can exhibit more complex behaviors, especially at high pressures or low temperatures.

  2. Constant Volume: The cylinder's volume is constant because the piston is fixed. This means V₁ = V₂. Constant volume is crucial because it simplifies the energy balance. In a constant volume process, all the heat added to the system goes into increasing its internal energy, which directly translates to an increase in temperature. If the volume were to change, some of the energy would be used to do work (like pushing the piston), making the analysis more complex.

  3. Electrical Energy to Heat: We'll calculate the electrical energy supplied by the heater and assume it's entirely converted into heat absorbed by the gas. This is a simplification because, in reality, some energy might be lost to the surroundings as heat. However, for the purpose of this problem, we'll assume perfect efficiency. This involves using the formula E = IVt, where:

    • E = Energy
    • I = Current
    • V = Voltage
    • t = Time

    This conversion is a key step because it links the electrical input to the thermal response of the gas. By assuming all electrical energy becomes heat, we can directly calculate the energy input and its effect on the gas's temperature.

  4. Specific Heat: We'll need the specific heat of nitrogen at constant volume (Cv) to relate the heat added to the temperature change. This value tells us how much energy is required to raise the temperature of one mole of nitrogen by one degree Celsius at constant volume. The specific heat capacity is essential for determining how much the temperature of the gas will rise for a given amount of heat input. Different gases have different specific heat capacities, and it’s important to use the correct value for nitrogen to get accurate results.

By making these assumptions, we create a simplified model that allows us to tackle the problem using basic thermodynamic principles. It's like setting up a controlled experiment where we can isolate the key variables and understand their interactions.

Calculating the Heat Input

The first thing we need to figure out is how much heat the electric heater is pumping into the nitrogen gas. Remember, we're assuming all the electrical energy gets turned into heat. The formula for electrical energy is:

  • E = IVt

But wait! We don't have the voltage (V). We're given the current (I) and time (t), but we need voltage to calculate the energy directly. Hmmm, what do we do? This is a common trick in these types of problems. Instead of voltage, we can consider the power (P) of the heater and use the formula:

  • E = Pt

And the power can be expressed as:

  • P = IV

However, without the voltage, we can’t directly use P = IV. We need to think a bit outside the box here. Sometimes, problem statements assume a standard voltage, or there might be a hidden piece of information. For now, let's continue as if we had the power supplied directly. If we circle back and realize we need to estimate or look up a typical voltage, we can do that later. For the sake of moving forward, let’s assume we have the power P (in Watts) from the heater. The time t is given as 4.8 minutes, which we need to convert to seconds:

  • t = 4.8 minutes * 60 seconds/minute = 288 seconds

So, the electrical energy (which we're assuming is the heat input) is:

  • Q = E = P * 288 seconds

The Importance of Units

It's super important, guys, to keep track of your units! We're dealing with different units here: kilopascals (kPa) for pressure, degrees Celsius (°C) for temperature, amps (A) for current, minutes and seconds for time, and we’ll likely end up with Joules (J) for energy and heat. Making sure everything is in the correct units (SI units are usually best) will prevent a lot of headaches later on. Converting minutes to seconds was a small but crucial step in ensuring our calculations will be accurate.

Dealing with Missing Information

You might be thinking,