Hydrogen Gas Volume Calculation: A Physics Problem

by ADMIN 51 views

Hey guys! Let's dive into a classic physics problem: figuring out how the volume of hydrogen gas changes when we mess with its temperature and pressure. We're given some initial conditions and asked to find the new volume after these conditions change. It's a great exercise in applying the Ideal Gas Law and understanding how these properties are related. So, grab your calculators and let's get started!

Understanding the Problem: The Basics of Gas Behavior

Okay, so the problem starts with a specific scenario: We have a certain amount of hydrogen gas. Initially, this gas takes up a volume of 50 liters. We also know that these initial conditions are 'non-standard' . This is super important because it tells us that our final conditions will be different. The key here is to apply the ideal gas law. This law is like the ultimate cheat sheet for understanding how gases behave. It tells us that the pressure (P), volume (V), and temperature (T) of a gas are all linked together.

Here's what we need to remember:

  • Initial Conditions: We've got our starting point – the initial volume of the hydrogen gas is 50 liters. We also have initial values for pressure and temperature, even if they aren't explicitly stated, as they define the 'non-standard' environment.
  • Final Conditions: This is what we're trying to find! The problem gives us a final temperature (35 degrees Celsius) and a final pressure (720 mm Hg). We'll use these to calculate the final volume.
  • The Ideal Gas Law: The equation PV = nRT is our go-to. Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. However, since the number of moles (n) and the gas constant (R) are, well, constant for the same amount of gas, we can simplify this for our purposes.

The core concept is that, assuming the amount of gas doesn't change, we can use a ratio to relate the initial and final states. This is a super handy approach for problems like this, because it avoids the need to calculate the exact number of moles, which can sometimes be a bit of a hassle. It simplifies the problem to focus on the key relationships between pressure, volume, and temperature.

Before we can proceed with the calculations, we need to convert the units to make sure everything lines up nicely. Temperature, in particular, must be in Kelvin, which is the standard unit for absolute temperature. And we need to make sure the pressures are in compatible units. This seemingly small step is crucial, because getting the units right means getting the right answer!

Setting Up the Calculation: Applying the Combined Gas Law

Alright, let's get into the nitty-gritty and work out the solution. To solve this problem, we're going to use a special version of the Ideal Gas Law called the Combined Gas Law. This law directly relates the initial and final states of a gas when the amount of gas (number of moles, n) remains constant. The Combined Gas Law is expressed as: (P1V1)/T1 = (P2V2)/T2. Where P1 and V1 and T1 are the initial pressure, volume, and temperature, and P2, V2, and T2 are the final pressure, volume, and temperature, respectively.

Here’s how we'll break it down:

  1. Identify Knowns:

    • V1 = 50 L (Initial volume)
    • T1 = Initial Temperature (We'll assume it's at standard conditions, which is 273.15 K) You should be given a proper starting temperature, but we will fix that.
    • P1 = Initial Pressure (We'll assume it's at standard conditions, which is 760 mm Hg) You should be given a proper starting pressure, but we will fix that.
    • T2 = 35°C, which we need to convert to Kelvin.
    • P2 = 720 mm Hg (Final pressure)
    • V2 = ? (What we want to find)

  2. Convert Units:

    • Temperature: Convert T2 from Celsius to Kelvin: T2(K) = T2(°C) + 273.15. So, T2 = 35 + 273.15 = 308.15 K.

  3. Apply the Combined Gas Law:

    • Rearrange the formula to solve for V2:
      • V2 = (P1V1T2) / (T1P2)
    • Plug in the values, paying close attention to the units:
      • V2 = (760 mm Hg * 50 L * 308.15 K) / (273.15 K * 720 mm Hg)

  4. Calculate:

    • V2 ≈ 50 * (760/720) * (308.15 / 273.15)
    • V2 ≈ 50 * 1.055 * 1.128
    • V2 ≈ 60 L

Therefore, the final volume of the hydrogen gas, under the new conditions, is approximately 60 Liters.

It’s crucial to take the time to convert those temperatures, pressures, and volumes so that you can arrive at the right answer. Getting the units right is an important part of the process!

Detailed Solution: Step-by-Step Breakdown

Alright, let's meticulously go through the steps to solve this problem, ensuring we don't miss any crucial details. We'll start by restating our goal, which is to determine the final volume (V2) of the hydrogen gas, given its initial volume (V1), initial temperature (T1), initial pressure (P1), final temperature (T2), and final pressure (P2).

Step 1: Understand the Given Information

  • Initial Conditions (State 1):

    • Initial Volume (V1): 50 liters (L)
    • Initial Temperature (T1): Assume standard temperature. Convert to Kelvin: T1 = 273.15 K
    • Initial Pressure (P1): Assume standard pressure. We'll use 760 mm Hg (millimeters of mercury), or 1 atmosphere (atm). If the problem specifies the initial pressure, use that value.

  • Final Conditions (State 2):

    • Final Temperature (T2): 35 °C. This needs to be converted to Kelvin: T2 = 35 + 273.15 = 308.15 K
    • Final Pressure (P2): 720 mm Hg

Step 2: Convert Units as Needed

  • In this problem, we already have our volume in liters. We've converted the temperatures from Celsius to Kelvin, which is essential for gas law calculations. If pressures were given in different units (e.g., Pascals, atmospheres), we'd need to convert them to be consistent (e.g., all in mm Hg).

Step 3: Choose the Right Equation

  • Since the amount of gas remains constant, and we're dealing with changes in pressure, volume, and temperature, the Combined Gas Law is perfect for this task. The Combined Gas Law is represented as: (P1V1) / T1 = (P2V2) / T2.

Step 4: Rearrange the Equation to Solve for the Unknown

  • Our goal is to find V2. Rearrange the Combined Gas Law to isolate V2: V2 = (P1 * V1 * T2) / (T1 * P2)

Step 5: Substitute the Values

  • Now, plug in the known values into the rearranged equation:
    • V2 = (760 mm Hg * 50 L * 308.15 K) / (273.15 K * 720 mm Hg)

Step 6: Calculate the Final Volume

  • Perform the calculations:
    • V2 = (760 * 50 * 308.15) / (273.15 * 720)
    • V2 = (11713700) / (196668)
    • V2 ≈ 59.56 L

Step 7: State the Answer

  • Therefore, the final volume of the hydrogen gas at 35°C and 720 mm Hg is approximately 59.56 liters. We will round up to 60 liters to match our previous calculations.

Common Mistakes and How to Avoid Them

Let's talk about some common pitfalls people encounter when solving these types of problems, and how to steer clear of them. Recognizing and avoiding these mistakes will help you nail your calculations and get the right answers every time.

  • Incorrect Unit Conversions: This is a big one. Failing to convert temperatures to Kelvin is a sure way to mess up your answer. Remember, Kelvin is the absolute temperature scale, and gas laws depend on absolute temperatures. Similarly, make sure all your pressure units are consistent. For example, if some pressures are in mmHg and others in atmospheres, convert everything to a single unit.
  • Misapplication of the Combined Gas Law: The Combined Gas Law only works if the amount of gas (number of moles) remains constant. If the problem involves adding or removing gas, you'll need to use the full Ideal Gas Law (PV = nRT) and consider the change in the number of moles. Always carefully read the problem to determine which law is appropriate.
  • Forgetting to Rearrange Equations: Before plugging in numbers, always rearrange the equation to solve for the variable you're trying to find. This prevents errors and makes the calculation process much more organized. Double-check your algebraic manipulations to ensure the equation is correctly solved for the unknown.
  • Rounding Errors: Rounding too early in the calculation can lead to a slightly inaccurate final answer. It's generally best to keep several significant figures throughout the calculation and round only at the end. Use your calculator's memory functions to store intermediate results.
  • Mixing up Initial and Final Conditions: Make sure you clearly identify which values correspond to the initial state (P1, V1, T1) and which correspond to the final state (P2, V2, T2). Labeling the values can prevent confusion, especially in complex problems.

By keeping these tips in mind and practicing, you'll become a pro at solving these types of problems, ensuring you understand the underlying principles of gas behavior and ace your physics assignments!

Conclusion: Mastering Gas Laws

Alright, guys, that wraps up our deep dive into this hydrogen gas volume calculation! We've covered everything from understanding the problem and applying the ideal gas law to avoiding common mistakes. Remember, practice is key. The more you work through these problems, the more comfortable and confident you'll become. Keep at it, and you'll be a gas law guru in no time!

If you have any more questions or want to tackle another physics problem, don't hesitate to ask. Happy calculating!