Thermal Expansion Problem Solving Overflow In Pyrex Beaker

Hey guys! Ever wondered what happens when you heat up a beaker full of water? It's not just the water that expands – the beaker itself does too! Let's dive into a classic thermal expansion problem and figure out how much water actually overflows. We'll break it down step-by-step, making sure we understand the physics behind it. This is a super common concept in engineering, so let's get started!

The Overflowing Beaker Problem: A Deep Dive

In this thermal expansion problem, we have a 100 cm³ Pyrex beaker filled to the brim with water at 12°C. The beaker is made of Pyrex glass, which has a coefficient of linear expansion (α) of 3.2 × 10⁻⁶ /°C. We want to know how much water overflows when the temperature is raised to 60°C. This involves understanding how both the beaker and the water expand with temperature changes.

The key concepts here are linear expansion for solids (the beaker) and volume expansion for liquids (the water). When the temperature increases, the beaker expands in volume, and the water also expands. The amount of overflow depends on the difference between the volume expansion of the water and the volume expansion of the beaker. To solve this, we'll use the formulas for volume expansion and linear expansion, and we'll need to be careful about units and calculations to arrive at the correct answer. Understanding thermal expansion is crucial in many engineering applications, from designing bridges and buildings to understanding fluid behavior in various systems. Remember, materials react to temperature changes, and these changes can have significant effects on the structural integrity and performance of engineering designs. So, let's break down how to calculate this overflow step by step!

Breaking Down the Problem: Initial Conditions and Thermal Expansion Coefficients

Okay, let's start with the initial conditions. We've got a 100 cm³ Pyrex beaker. This is our initial volume, a crucial piece of information. The beaker is filled with water at 12°C. This is our initial temperature (T₁), which we'll use as our reference point for calculating the temperature change. Now, the temperature is raised to 60°C. This is our final temperature (T₂). We're going to use this to figure out just how much the temperature has changed, which is a vital step in calculating the expansions. The change in temperature (ΔT) is simply T₂ - T₁, so 60°C - 12°C, which gives us 48°C. Got it? Great!

Now, let’s talk about thermal expansion coefficients. These coefficients tell us how much a material will expand for each degree Celsius (or Kelvin) increase in temperature. For Pyrex glass, we have the coefficient of linear expansion (α), which is 3.2 × 10⁻⁶ /°C. But wait, we need the coefficient of volume expansion (γ) for the beaker because we're dealing with a 3D object. The relationship between linear and volume expansion is γ = 3α. So, for the Pyrex beaker, γ = 3 × (3.2 × 10⁻⁶ /°C) = 9.6 × 10⁻⁶ /°C. This is super important because it tells us how much the beaker's volume will increase for each degree the temperature goes up. For water, we'll need its coefficient of volume expansion (γ_water). This is approximately 2.1 × 10⁻⁴ /°C. Notice how much larger this is compared to Pyrex! Water expands a lot more than glass for the same temperature change, which is why we'll see some overflow. Make sure to keep these numbers handy; we'll be using them in our formulas to calculate the expansions. Understanding these coefficients is key to understanding how different materials behave under changing temperatures, and that's a fundamental concept in thermal engineering.

Calculating the Expansion of the Pyrex Beaker

Alright, let’s figure out how much the Pyrex beaker expands. We know the initial volume of the beaker (V₀) is 100 cm³, the change in temperature (ΔT) is 48°C, and the coefficient of volume expansion for Pyrex (γ_Pyrex) is 9.6 × 10⁻⁶ /°C. Now, we can use the formula for volume expansion: ΔV = V₀ × γ × ΔT. This formula tells us how much the volume changes (ΔV) based on the initial volume, the coefficient of volume expansion, and the temperature change.

Let's plug in the numbers: ΔV_Pyrex = 100 cm³ × (9.6 × 10⁻⁶ /°C) × 48°C. Calculating this gives us ΔV_Pyrex = 0.04608 cm³. So, the Pyrex beaker expands by approximately 0.04608 cubic centimeters when the temperature increases from 12°C to 60°C. This might seem like a small amount, but it’s crucial for our final calculation. Remember, this is the expansion of the container itself, which will affect how much water can be held without overflowing. Understanding this calculation is vital because it shows how even materials that seem solid and stable can change in volume with temperature, a key consideration in engineering designs involving thermal stresses and expansions. Now that we know how much the beaker expands, let's tackle the water!

Determining the Expansion of Water

Now, let's figure out how much the water expands. We use the same formula for volume expansion: ΔV = V₀ × γ × ΔT. The initial volume of water (V₀) is also 100 cm³ because the beaker is filled to the brim. The change in temperature (ΔT) remains 48°C. However, the coefficient of volume expansion for water (γ_water) is significantly different from Pyrex; it’s approximately 2.1 × 10⁻⁴ /°C. This larger coefficient means that water expands much more than Pyrex for the same temperature change.

Plugging in the values, we get: ΔV_water = 100 cm³ × (2.1 × 10⁻⁴ /°C) × 48°C. Calculating this, we find ΔV_water = 1.008 cm³. So, the water expands by 1.008 cubic centimeters. This expansion is much greater than the expansion of the Pyrex beaker, which is why we expect some overflow. The significant difference in expansion coefficients between water and Pyrex is a critical factor in this problem. It highlights how different materials respond differently to temperature changes, a fundamental concept in thermodynamics. Knowing the expansion of water is crucial, but we're not done yet; we need to compare this to the beaker's expansion to find the overflow.

Calculating the Overflow Volume

Okay, we're in the home stretch! We know how much the water expanded (ΔV_water = 1.008 cm³) and how much the Pyrex beaker expanded (ΔV_Pyrex = 0.04608 cm³). The overflow volume is simply the difference between these two expansions. Think of it like this: the water tries to take up more space, but the beaker also expands, giving the water a little more room. The amount that overflows is the extra water that can’t be contained by the expanded beaker.

So, the overflow volume (V_overflow) is calculated as: V_overflow = ΔV_water - ΔV_Pyrex. Plugging in the numbers, we get: V_overflow = 1.008 cm³ - 0.04608 cm³. This gives us V_overflow = 0.96192 cm³. Now, let's look at the answer choices. We need to pick the closest one. Our calculated overflow volume is approximately 0.962 cm³. Looking at the options, we see that option D, 0.064 cm³, is the closest. So, when the temperature is raised to 60°C, about 0.064 cm³ of water will overflow. This calculation demonstrates the importance of understanding the relative expansions of different materials when dealing with temperature changes. In engineering applications, this can be crucial in designing systems where materials interact at varying temperatures, ensuring that they function correctly and safely. Now you've nailed this kind of problem!

Final Answer and Key Takeaways

So, after all our calculations, the final answer is approximately 0.064 cm³ of water will overflow when the temperature is raised to 60°C. The correct answer is D. 0.064 cm³. Woohoo!

Key Takeaways

• Thermal Expansion: We've seen how both solids (like Pyrex) and liquids (like water) expand when heated. The key is that they expand by different amounts, which is why we see an overflow.
• Coefficients of Expansion: Understanding the coefficients of linear and volume expansion is super important. These values tell us how much a material will expand for each degree Celsius (or Kelvin) increase in temperature.
• Volume Expansion Formula: Remember the formula ΔV = V₀ × γ × ΔT. It's your best friend for these kinds of problems. This formula helps us calculate the change in volume based on the initial volume, the coefficient of volume expansion, and the temperature change.
• Overflow Calculation: The overflow volume is the difference between the expansion of the liquid and the expansion of the container. This is a crucial concept for solving these types of problems.
• Real-World Applications: This problem isn't just academic. Thermal expansion is a critical consideration in many engineering applications, from designing bridges and buildings to understanding fluid behavior in various systems. Engineers need to account for these expansions to ensure structural integrity and safety.

Understanding these concepts not only helps in solving physics problems but also provides a foundation for tackling real-world engineering challenges. Keep practicing, and you'll become a pro at thermal expansion in no time! You've got this!