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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
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| <title>The gas thermometer consists in two identical containers with volume V_0 each one, linked by a tube of length l and cross section S. A mercury droplet obstructs the tube. If temperatures of gasses in the volumes are equal, the mercury lies in the center of the tube. The right volume is connected to a thermostat with temperature T_0. Scale the thermometer finding the dependence between gas temperature in left volume and the displacement x of mercury respect to equilibrium position.</title> | | <title>The gas thermometer consists in two identical containers with volume V_0 each one, linked by a tube of length l and cross section S. A mercury droplet obstructs the tube. If temperatures of gasses in the volumes are equal, the mercury lies in the center of the tube. The right volume is connected to a thermostat with temperature T_0. Scale the thermometer finding the dependence between gas temperature in left volume and the displacement x of mercury respect to equilibrium position.</title> |
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| <h3 id="back-link"><a href="../../#5.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#5.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| | | |
| <p> | | <p> |
| $5.5.16$ | | $5.5.16$ |
| The gas thermometer consists in two identical containers with volume $V_0$ each one, linked by a tube of length $l$ and cross section $S$. A mercury droplet obstructs the tube. If temperatures of gasses in the volumes are equal, the mercury lies in the center of the tube. The right volume is connected to a thermostat with temperature $T_0$. Scale the thermometer finding the dependence between gas temperature in left volume and the displacement $x$ of mercury respect to equilibrium position. | | The gas thermometer consists in two identical containers with volume $V_0$ each one, linked by a tube of length $l$ and cross section $S$. A mercury droplet obstructs the tube. If temperatures of gasses in the volumes are equal, the mercury lies in the center of the tube. The right volume is connected to a thermostat with temperature $T_0$. Scale the thermometer finding the dependence between gas temperature in left volume and the displacement $x$ of mercury respect to equilibrium position. |
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| loading="lazy" alt="5.5.16" width="250" /> | | loading="lazy" alt="5.5.16" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem 5.5.16 | | For problem 5.5.16 |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container | | Let denote left volume as $V_1$ and right one as $V_2$. Applying state equation for each container |
| $$P_1~V_1 = nRT \;(1)$$ | | $$P_1~V_1 = nRT \;(1)$$ |
| $$P_0~V_2 = nRT_0 \;(2)$$ | | $$P_0~V_2 = nRT_0 \;(2)$$ |
| It is easy to see that | | It is easy to see that |
| $$V_1 = V_0+S\frac{l}{2}-Sx \;(3)$$ | | $$V_1 = V_0+S\frac{l}{2}-Sx \;(3)$$ |
| $$V_2 = V_0+S\frac{l}{2}+Sx \;(4)$$ | | $$V_2 = V_0+S\frac{l}{2}+Sx \;(4)$$ |
| If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$ | | If droplet is in equilibrium that means pressure forces due to gasses are equal, so $P_1 = P_0$. Then, dividing $(1)$ and $(2)$ |
| $$\frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$$ | | $$\frac{V_1}{V_2}=\frac{T}{T_0} \;(5)$$ |
| Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$ | | Substituting $(3)$ and $(4)$ into $(5)$ and separating $T$ |
| $$\boxed{T = T_0~\frac{2V_0+S(l+2x)}{2V_0+S(l-2x)}}$$ | | $$\boxed{T = T_0~\frac{2V_0+S(l+2x)}{2V_0+S(l-2x)}}$$ |
| This is mathematically valid for $0< x <\frac{V_0}{S}+\frac{l}{2}$ but according the physical situation only for $0< x < \frac{l}{2}$. | | This is mathematically valid for $0< x <\frac{V_0}{S}+\frac{l}{2}$ but according the physical situation only for $0< x < \frac{l}{2}$. |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
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