热学

$m_{\text{partical}}\overset {N_A} \longleftrightarrow m_{\text{mol}}\overset{n}{\longleftrightarrow}m$ $\begin{gathered} \hfill \\ \left\{ \begin{gathered} {p_1}{V_1} = {p_2}{V_2} \hfill \\ pV = C \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered}$

$\begin{array}{|c|c|} \hline &p&V\\ \hline 原& p_1V_{1}+p_2V_{2}&\backslash \\ \hline 后&p&V_1+V_2\\ \hline \end{array}$ $p_1V_1+p_2V_2=p(V _1+V_2)$

$\frac{4}{3}\pi {r^3} = \frac{1}{6}\pi {d^3} = \frac{m}{n \cdot {N_A} \cdot \rho } \Rightarrow d = \sqrt[3]{\frac{6m}{n{N_A}\pi \rho }}$ $\left\{ \begin{gathered} solid \to V = \frac{ m_{partical} } {\rho } = \frac{m} { n\rho {N_A} } \hfill \\ gas \to V = \frac{ m_{partical} } {\rho }(\frac{m} { n\rho {N_A} } = distance) \hfill \\ \end{gathered} \right.$

$S_Ap_A+p_0S_B=mg\sin\theta+S_Bp_B+p_0S_A$

打气问题

$\sum\limits_{i = 1}^n {\frac{p_iV_i}{T_i}} = \frac{pV}{T}=nR$ $p_0V_0+p_0\triangle V =p_1V_0\\ p_1V_0+p_0\triangle V=p_2V_0\\ \cdots\\ p_{n-1}V_0+p_0\triangle V=p_nV_0 \\\Rightarrow p_n=\left(1+\frac{n\triangle V}{V_0}\right)p_0$

抽气问题

$p_0V_0=p_1(V_0+\triangle V)\\ p_1V_0=p_2(V_0+\triangle V)\\ \cdots\\ p_{n-1}V_0=p_n(V_0+\triangle V)\\ \Rightarrow p_n=\left(\frac{\triangle V}{\triangle V+V_0}\right)^np_0$ $mg\sin \theta - \mu (Eq + mg\cos \theta ) = ma$