Nitrogen Gas In A Cylinder Heated Electrically Problem And Solution

October 10, 2025 by Scholario Team 68 views

Hey guys! Ever wondered what happens when you heat nitrogen gas in a cylinder? Well, let's dive into a super interesting problem involving nitrogen gas, a cylinder, and a heater. We'll break down the problem step by step, making it super easy to understand. Let's get started!

Problem Statement: Unpacking the Nitrogen Cylinder Scenario

Okay, so here’s the deal. We've got some nitrogen gas chilling inside a cylinder that’s equipped with a piston. The gas is currently at a pressure of 398 kPa and a temperature of 26.8°C. Now, inside this cylinder, there’s an electric heater. This heater gets turned on, and it’s passing a current of 1.9 A for 4.8 minutes using a 120 V power source. This setup introduces some really cool physics concepts that we can explore.

Initial Conditions: Setting the Stage

First, let's highlight the initial conditions because they're super important for solving this kind of problem. We know:

Pressure (P₁): 398 kPa
Temperature (T₁): 26.8°C
Current (I): 1.9 A
Time (t): 4.8 minutes
Voltage (V): 120 V

These numbers are like the opening scene of our physics movie – they set the stage for what’s about to happen. Understanding these initial conditions is crucial because they'll affect everything that follows. Think of it like baking a cake; if you don't know the starting temperature and the ingredients, you're gonna have a hard time getting the right result!

The Heater's Role: Adding Energy to the System

Now, let’s talk about the heater. This little guy is the game-changer in our scenario. When the heater is turned on, it starts pumping energy into the system. It’s like adding fuel to a fire, but in this case, it’s electrical energy turning into heat. The key here is that this heat is going to affect the nitrogen gas inside the cylinder.

The heater operates at 120 V and passes 1.9 A of current. This electrical energy is converted into thermal energy, which will increase the gas's internal energy and, potentially, its volume and temperature. The time the heater operates, 4.8 minutes, is also crucial because it tells us how long this energy transfer is happening. It's like knowing how long you're going to bake that cake for – too short, and it’s raw; too long, and it’s burnt. The same goes for energy transfer!

What We Want to Know: The Million-Dollar Question

So, what's the big question here? What are we trying to figure out? Well, typically, in problems like these, we might want to know things like:

How much heat was added to the system?
What's the final temperature of the gas?
Did the volume of the gas change, and if so, by how much?

To answer these questions, we'll need to use the principles of thermodynamics – things like the ideal gas law, the first law of thermodynamics, and maybe some specific heat calculations. Don’t worry if these sound scary; we'll break them down step by step. It's like following a recipe; each step gets you closer to the final delicious result!

In summary, the problem gives us a scenario with nitrogen gas in a cylinder, heated by an electric heater. We know the initial conditions and the heater's electrical parameters. Now, we need to put on our thinking caps and figure out how to use this information to solve for the unknowns. Let's dive into the solution next!

Step-by-Step Solution: Cracking the Code

Alright, let’s get to the fun part – actually solving the problem! We're going to take it one step at a time, so you can see exactly how we get to the answer. Remember, it's like solving a puzzle; each piece fits together to give you the big picture.

Step 1: Calculate the Electrical Energy Input

The first thing we need to figure out is how much energy the heater is pumping into the system. We know the voltage, current, and time, so we can use a simple formula from physics to find the electrical energy (E).

The formula is:

E = V × I × t

Where:

E is the energy in Joules (J)
V is the voltage in Volts (V)
I is the current in Amperes (A)
t is the time in seconds (s)

But wait! Our time is in minutes, so we need to convert that to seconds. There are 60 seconds in a minute, so:

t = 4.8 minutes × 60 seconds/minute = 288 seconds

Now we can plug in the values:

E = 120 V × 1.9 A × 288 s E = 65664 J

So, the heater adds 65664 Joules of energy to the system. That’s like a significant burst of energy! This value is crucial because it tells us how much heat is being added to the gas, which will then affect its temperature and pressure.

Step 2: Understanding the Thermodynamic Process

Before we jump into more calculations, let’s take a moment to think about what’s happening inside the cylinder. The nitrogen gas is being heated, but what kind of process is this? Is the volume constant? Is the pressure constant? This is where our understanding of thermodynamics comes in handy.

In this scenario, the cylinder has a piston. This detail is super important because it tells us that the piston can move. If the piston can move, the gas can expand. Usually, we assume that if a piston is free to move, the pressure inside the cylinder remains constant (assuming the external pressure is constant). This is called an isobaric process.

Isobaric might sound like a fancy word, but it just means “constant pressure.” So, as the gas heats up, it expands, pushing the piston outward, but the pressure stays the same. It’s like blowing up a balloon; you’re adding more air (in our case, energy), and the balloon gets bigger, but the pressure inside stays relatively constant.

Knowing this is an isobaric process simplifies our calculations because we can use equations that apply specifically to constant pressure scenarios. This is a critical step in problem-solving – identifying the type of process helps you choose the right tools (or equations) for the job.

Step 3: Applying the First Law of Thermodynamics

Now that we know the energy input and the type of process, we can use the First Law of Thermodynamics to relate the heat added to the changes in the system. The First Law is basically a statement of energy conservation: energy can't be created or destroyed, only converted from one form to another.

The First Law of Thermodynamics can be written as:

ΔU = Q - W

Where:

ΔU is the change in internal energy of the gas
Q is the heat added to the gas
W is the work done by the gas

In our case, the heat added (Q) is the electrical energy we calculated in Step 1, which is 65664 J. Now we need to figure out the work done by the gas (W) and the change in internal energy (ΔU).

For an isobaric process, the work done can be calculated as:

W = P × ΔV

Where:

P is the constant pressure
ΔV is the change in volume

So, we need to find the change in volume to calculate the work done. This is where we might need more information, such as the initial volume or the number of moles of gas. If we had that, we could use the ideal gas law to find the final volume and then calculate ΔV.

Step 4: Potential Next Steps and Required Information

Okay, so we’ve made some good progress. We’ve calculated the heat added, identified the process as isobaric, and set up the First Law of Thermodynamics. But, we’ve hit a bit of a roadblock. To continue, we need more information.

Specifically, we need either:

The initial volume of the nitrogen gas.
The number of moles of nitrogen gas.

If we had either of these, we could use the ideal gas law to find the change in volume and then calculate the work done. The ideal gas law is:

PV = nRT

Where:

P is the pressure
V is the volume
n is the number of moles
R is the ideal gas constant
T is the temperature

Once we know the work done, we can go back to the First Law of Thermodynamics and find the change in internal energy (ΔU). From there, we could calculate the final temperature if we knew the specific heat of nitrogen gas.

So, in summary, we’ve taken the problem step by step, but we’ve identified that we need more information to get to the final answer. It's like being a detective; you've gathered some clues, but you need a few more to crack the case!

Conclusion: Tying It All Together

So, guys, we've taken a deep dive into a problem involving nitrogen gas in a cylinder heated by an electric heater. We started by understanding the initial conditions and the role of the heater in adding energy to the system. We calculated the electrical energy input and identified the process as isobaric, which means the pressure remains constant.

We then applied the First Law of Thermodynamics to relate the heat added to the changes in the system. However, we hit a point where we realized we needed more information – either the initial volume or the number of moles of nitrogen gas – to proceed further. This is a common situation in problem-solving; sometimes you need to go back and gather more data before you can complete the solution.

This problem highlights the importance of understanding key concepts like:

Energy Conservation: The First Law of Thermodynamics is all about energy conservation.
Thermodynamic Processes: Identifying the type of process (isobaric, isothermal, etc.) is crucial for choosing the right equations.
The Ideal Gas Law: A fundamental tool for relating pressure, volume, temperature, and the number of moles of a gas.

Even though we couldn’t reach a final numerical answer without more information, we’ve demonstrated the process of breaking down a complex problem into manageable steps. We’ve used physics principles to set up the solution, and we’ve identified exactly what information we need to proceed.

Remember, in physics (and in life!), it’s not always about getting the final answer right away. It’s about understanding the process, asking the right questions, and knowing what tools you have at your disposal. So, next time you see a problem like this, you’ll be ready to tackle it step by step!

I hope you found this breakdown helpful. Keep exploring, keep questioning, and keep learning! Until next time!