Collection of Solved Problems in Physics

Laboratory Problem

Task number: 1291

• Hint

  The volume of the water will increase with the increasing temperature, but the graduated cylinder will also expand. Our task is to find out which volume increases more, whether the water volume or the inner volume of the graduated cylinder.

• How to determine how the inside of the cylinder increases?

  The formula for thermal expansion tells us how the body enlarges. But can it be applied to "the inside of the cylinder", i.e. the cavity inside the body?

  Yes, we can use it. Just imagine that we would fill the graduated cylinder with a glass cylinder. Then we would warm up both the graduated cylinder and the glass cylinder. Because they are made of the same material, the expansion would be the same. This means that the enlarged inside of the graduated cylinder exactly corresponds to the increased volume of the glass cylinder, with which we had filled the graduated cylinder.

• Numerical values

  VC = 100 ml		the volume of water in the graduated cylinder
  t1 = 20 °C		the initial temperature
  t2 = 30 °C		the final temperature
  From The Handbook of Chemistry and Physics
  βw = 0.2·10−3K−1		the coefficient of volumetric thermal expansion of water
  αg = 8·10−6K−1		tthe coefficient of linear thermal expansion of glass

• Analysis

  At first glance, it might seem that the task has a completely clear solution. When the temperature increases, the water volume also increases, so the water level will be above the line. However, on closer examination we will realize that not only the water but also the glass expands with the increasing temperature, so when the temperature increases, the inner volume of the graduated cylinder also increases. If the expansion of glass outweighed the expansion of water, the water level would be below the line. Our task is therefore to compare these two contradictory effects.

• Solution

  First we will determine the volume of water at the higher temperature. For the volumetric expansion of water the following formula is true

  \[V_w = V_C (1 + \beta_W\Delta t).\]

  Furthermore, we will determine the internal volume of the graduated cylinder at the higher temperature. In the Handbook of Chemistry and Physics, however, there is only the linear expansion coefficient listed. We will therefore have to use the formula for the volume of the cylinder V = πr²h, where r is the radius of the cylinder and h is its height. We will mark the dimensions of the cylinder at the initial temperature with the same letters and differentiate them with index 0.

  We will now substitute the relations expressing the change in the dimensions with the increasing temperature into this formula:

  \[r = r_0 (1 + \alpha_g\Delta t)\] \[h = h_0 (1 + \alpha_g\Delta t)\] \[V_g=\pi r^2 h=\pi r_0^2(1+\alpha_g\Delta t)^2h_0(1+\alpha_g\Delta t)\] \[V_g=\pi r_0^2h_0(1+\alpha_g\Delta t)^3\] \[V_g=V_C(1+\alpha_g\Delta t)^3\]

  We will multiply out the parentheses and realize that the members with α with higher exponent will be in respect to α and Δt too small, and therefore can be neglected:

  \[V_g=V_C(1 + 3\alpha_g\Delta t+3\alpha_g^2\Delta^2 t+3\alpha_g^3\Delta t^3)\]

  The last equation shows that with sufficient accuracy the following relationship is true between the thermal linear expansion coefficient α_g and the thermal volumetric expansion coefficient β_g:

  \[\beta_g\dot{=}3\alpha _g.\]

  Our task is therefore to compare the volumes expressed by the relations

  \[V_w = V_C (1 + \beta_w\Delta t),\] \[V_g = V_C (1 + 3\alpha_g\Delta t).\]

  It is true that β_W > 3 α_g, We can therefore say that the volume of water increases more than the inner volume of the graduated cylinder. The water level will therefore be above the line.

  Let us substitute the numerical values:

  \[V_w = V_C (1 + \beta_w\Delta t)=100\cdot[1+2\cdot{10^{-4}}\cdot(30-20)]\,\mathrm{ml}\] \[V_w = 100.2\,\mathrm{ml}\] \[V_g = V_C (1 + 3\alpha_g\Delta t)=100\cdot[1+3\cdot{8}\cdot{10^{-6}}\cdot(30-20)]\,\mathrm{ml}\] \[V_g=100.024\,\mathrm{ml}\]

  Now we can see that the internal volume of the graduated cylinder increases by a smaller value than the internal volume of water and the water will therefore be above the line indicating the volume of 100 ml.

• Answer

  After the temperature increase the water level will be above the line.

• Note – the relative thermal expansion

  The thermal expansion of the glass is very small (especially for certain types of glass such as silica glass), and can therefore be neglected in most situations.